Mister Exam
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- Graph of implicit function
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- Curve in 3D space defined parametrically
- Area between lines
- The intersection points of functions
-
How to use it?
- Graphing y =:
- x/(x^2-4x+3)
- x+cos2x
-
x^4-8x^3+16x^2
- x^4-4x^2+2
-
Identical expressions
- 19x- twelve /((5x- four)(x- thirteen))
- 19x minus 12 divide by ((5x minus 4)(x minus 13))
- 19x minus twelve divide by ((5x minus four)(x minus thirteen))
- 19x-12/5x-4x-13
- 19x-12 divide by ((5x-4)(x-13))
-
Similar expressions
- 19x+12/((5x-4)(x-13))
- 19x-12/((5x-4)(x+13))
- 19x-12/((5x+4)(x-13))
Graphing y = 19x-12/((5x-4)(x-13))
The solution
You have entered
[src]
12
f(x) = 19*x - ------------------
(5*x - 4)*(x - 13)
$$f{\left(x \right)} = 19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)}$$
f = 19*x - 12*1/((x - 13)*(5*x - 4))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0.8$$
$$x_{2} = 13$$
$$x_{1} = 0.8$$
$$x_{2} = 13$$
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{23}{5} + \frac{1327}{75 \sqrt[3]{\frac{174513}{2375} + \frac{2 \sqrt{638655033} i}{4275}}} + \sqrt[3]{\frac{174513}{2375} + \frac{2 \sqrt{638655033} i}{4275}}$$
so we need to solve the equation:
$$19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{23}{5} + \frac{1327}{75 \sqrt[3]{\frac{174513}{2375} + \frac{2 \sqrt{638655033} i}{4275}}} + \sqrt[3]{\frac{174513}{2375} + \frac{2 \sqrt{638655033} i}{4275}}$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- \frac{12 \left(69 - 10 x\right)}{\left(x - 13\right)^{2} \left(5 x - 4\right)^{2}} + 19 = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}$$
$$x_{2} = - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}$$
The values of the extrema at the points:
Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
$$x_{1} = - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}$$
Maxima of the function at points:
$$x_{1} = - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}$$
Decreasing at intervals
$$\left(-\infty, - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}\right] \cup \left[- \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}, \infty\right)$$
Increasing at intervals
$$\left[- \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}, - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$- \frac{12 \left(69 - 10 x\right)}{\left(x - 13\right)^{2} \left(5 x - 4\right)^{2}} + 19 = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}$$
$$x_{2} = - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}$$
The values of the extrema at the points:
________________________________________________________________________________________________________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________________________________________________________________________________________________________
/ _____________________________________ / _____________________________________
/ / ________________ / / ________________
/ 7442 / 18598860004321 \/ 55796572722963 13845841 48 / 7442 / 18598860004321 \/ 55796572722963 13845841 48
/ ---- - 2*3 / -------------- + ------------------ - ------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------- 19* / ---- - 2*3 / -------------- + ------------------ - ------------------------------------------------ + ---------------------------------------------------------------------------------------------------------------------
________________________________________________________________________________________________________ / 75 \/ 1218375000 676875 _____________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ / 75 \/ 1218375000 676875 _____________________________________ ________________________________________________________________________________________________________
/ _____________________________________ / / ________________ / _____________________________________ / _____________________________________ / / ________________ / _____________________________________
/ / ________________ / / 18598860004321 \/ 55796572722963 / / ________________ / / ________________ / / 18598860004321 \/ 55796572722963 / / ________________
/ 3721 / 18598860004321 \/ 55796572722963 13845841 / 11250*3 / -------------- + ------------------ / 3721 / 18598860004321 \/ 55796572722963 13845841 / 3721 / 18598860004321 \/ 55796572722963 13845841 / 11250*3 / -------------- + ------------------ / 3721 / 18598860004321 \/ 55796572722963 13845841
/ ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ / \/ 1218375000 676875 95* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ 19* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ / \/ 1218375000 676875 95* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------
/ 75 \/ 1218375000 676875 _____________________________________ / / 75 \/ 1218375000 676875 _____________________________________ / 75 \/ 1218375000 676875 _____________________________________ / / 75 \/ 1218375000 676875 _____________________________________
/ / ________________ / / / ________________ / / ________________ / / / ________________
/ / 18598860004321 \/ 55796572722963 / / / 18598860004321 \/ 55796572722963 / / 18598860004321 \/ 55796572722963 / / / 18598860004321 \/ 55796572722963
/ 11250*3 / -------------- + ------------------ / / 11250*3 / -------------- + ------------------ / 11250*3 / -------------- + ------------------ / / 11250*3 / -------------- + ------------------
69 \/ \/ 1218375000 676875 \/ \/ \/ 1218375000 676875 1311 \/ \/ 1218375000 676875 \/ \/ \/ 1218375000 676875 12
(-- - ------------------------------------------------------------------------------------------------------------------ - -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------, ---- - --------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------)
10 2 2 10 2 2 / ________________________________________________________________________________________________________________________________________________________________________________________________________________________________\ / ________________________________________________________________________________________________________________________________________________________________________________________________________________________________\
| / _____________________________________ | | / _____________________________________ |
| / / ________________ | | / / ________________ |
| / 7442 / 18598860004321 \/ 55796572722963 13845841 48 | | / 7442 / 18598860004321 \/ 55796572722963 13845841 48 |
| / ---- - 2*3 / -------------- + ------------------ - ------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------- | | 5* / ---- - 2*3 / -------------- + ------------------ - ------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------- |
| ________________________________________________________________________________________________________ / 75 \/ 1218375000 676875 _____________________________________ ________________________________________________________________________________________________________ | | ________________________________________________________________________________________________________ / 75 \/ 1218375000 676875 _____________________________________ ________________________________________________________________________________________________________ |
| / _____________________________________ / / ________________ / _____________________________________ | | / _____________________________________ / / ________________ / _____________________________________ |
| / / ________________ / / 18598860004321 \/ 55796572722963 / / ________________ | | / / ________________ / / 18598860004321 \/ 55796572722963 / / ________________ |
| / 3721 / 18598860004321 \/ 55796572722963 13845841 / 11250*3 / -------------- + ------------------ / 3721 / 18598860004321 \/ 55796572722963 13845841 | | / 3721 / 18598860004321 \/ 55796572722963 13845841 / 11250*3 / -------------- + ------------------ / 3721 / 18598860004321 \/ 55796572722963 13845841 |
| / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ / \/ 1218375000 676875 95* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ | | 5* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ / \/ 1218375000 676875 95* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ |
| / 75 \/ 1218375000 676875 _____________________________________ / / 75 \/ 1218375000 676875 _____________________________________ | | / 75 \/ 1218375000 676875 _____________________________________ / / 75 \/ 1218375000 676875 _____________________________________ |
| / / ________________ / / / ________________ | | / / ________________ / / / ________________ |
| / / 18598860004321 \/ 55796572722963 / / / 18598860004321 \/ 55796572722963 | | / / 18598860004321 \/ 55796572722963 / / / 18598860004321 \/ 55796572722963 |
| / 11250*3 / -------------- + ------------------ / / 11250*3 / -------------- + ------------------ | | / 11250*3 / -------------- + ------------------ / / 11250*3 / -------------- + ------------------ |
| 61 \/ \/ 1218375000 676875 \/ \/ \/ 1218375000 676875 | |61 \/ \/ 1218375000 676875 \/ \/ \/ 1218375000 676875 |
|- -- - ------------------------------------------------------------------------------------------------------------------ - -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|-- - -------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
\ 10 2 2 / \2 2 2 / ________________________________________________________________________________________________________________________________________________________________________________________________________________________________ ________________________________________________________________________________________________________________________________________________________________________________________________________________________________
/ _____________________________________ / _____________________________________
/ / ________________ / / ________________
/ 7442 / 18598860004321 \/ 55796572722963 13845841 48 / 7442 / 18598860004321 \/ 55796572722963 13845841 48
/ ---- - 2*3 / -------------- + ------------------ - ------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------- 19* / ---- - 2*3 / -------------- + ------------------ - ------------------------------------------------ + ---------------------------------------------------------------------------------------------------------------------
/ 75 \/ 1218375000 676875 _____________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________ / 75 \/ 1218375000 676875 _____________________________________ ________________________________________________________________________________________________________
/ / ________________ / _____________________________________ / _____________________________________ / _____________________________________ / / ________________ / _____________________________________
/ / 18598860004321 \/ 55796572722963 / / ________________ / / ________________ / / ________________ / / 18598860004321 \/ 55796572722963 / / ________________
/ 11250*3 / -------------- + ------------------ / 3721 / 18598860004321 \/ 55796572722963 13845841 / 3721 / 18598860004321 \/ 55796572722963 13845841 / 3721 / 18598860004321 \/ 55796572722963 13845841 / 11250*3 / -------------- + ------------------ / 3721 / 18598860004321 \/ 55796572722963 13845841
/ \/ 1218375000 676875 95* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ 19* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ / \/ 1218375000 676875 95* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------
/ / 75 \/ 1218375000 676875 _____________________________________ / 75 \/ 1218375000 676875 _____________________________________ / 75 \/ 1218375000 676875 _____________________________________ / / 75 \/ 1218375000 676875 _____________________________________
/ / / ________________ / / ________________ / / ________________ / / / ________________
/ / / 18598860004321 \/ 55796572722963 / / 18598860004321 \/ 55796572722963 / / 18598860004321 \/ 55796572722963 / / / 18598860004321 \/ 55796572722963
/ / 11250*3 / -------------- + ------------------ / 11250*3 / -------------- + ------------------ / 11250*3 / -------------- + ------------------ / / 11250*3 / -------------- + ------------------
69 \/ \/ \/ 1218375000 676875 \/ \/ 1218375000 676875 1311 \/ \/ 1218375000 676875 \/ \/ \/ 1218375000 676875 12
(-- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------, ---- - --------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------)
10 2 2 10 2 2 / ________________________________________________________________________________________________________________________________________________________________________________________________________________________________ \ / ________________________________________________________________________________________________________________________________________________________________________________________________________________________________\
| / _____________________________________ | | / _____________________________________ |
| / / ________________ | | / / ________________ |
| / 7442 / 18598860004321 \/ 55796572722963 13845841 48 | | / 7442 / 18598860004321 \/ 55796572722963 13845841 48 |
| / ---- - 2*3 / -------------- + ------------------ - ------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------- | | 5* / ---- - 2*3 / -------------- + ------------------ - ------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------- |
| / 75 \/ 1218375000 676875 _____________________________________ ________________________________________________________________________________________________________ ________________________________________________________________________________________________________| | ________________________________________________________________________________________________________ / 75 \/ 1218375000 676875 _____________________________________ ________________________________________________________________________________________________________ |
| / / ________________ / _____________________________________ / _____________________________________ | | / _____________________________________ / / ________________ / _____________________________________ |
| / / 18598860004321 \/ 55796572722963 / / ________________ / / ________________ | | / / ________________ / / 18598860004321 \/ 55796572722963 / / ________________ |
| / 11250*3 / -------------- + ------------------ / 3721 / 18598860004321 \/ 55796572722963 13845841 / 3721 / 18598860004321 \/ 55796572722963 13845841 | | / 3721 / 18598860004321 \/ 55796572722963 13845841 / 11250*3 / -------------- + ------------------ / 3721 / 18598860004321 \/ 55796572722963 13845841 |
| / \/ 1218375000 676875 95* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ | | 5* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ / \/ 1218375000 676875 95* / ---- + 2*3 / -------------- + ------------------ + ------------------------------------------------ |
| / / 75 \/ 1218375000 676875 _____________________________________ / 75 \/ 1218375000 676875 _____________________________________ | | / 75 \/ 1218375000 676875 _____________________________________ / / 75 \/ 1218375000 676875 _____________________________________ |
| / / / ________________ / / ________________ | | / / ________________ / / / ________________ |
| / / / 18598860004321 \/ 55796572722963 / / 18598860004321 \/ 55796572722963 | | / / 18598860004321 \/ 55796572722963 / / / 18598860004321 \/ 55796572722963 |
| / / 11250*3 / -------------- + ------------------ / 11250*3 / -------------- + ------------------ | | / 11250*3 / -------------- + ------------------ / / 11250*3 / -------------- + ------------------ |
| 61 \/ \/ \/ 1218375000 676875 \/ \/ 1218375000 676875 | |61 \/ \/ 1218375000 676875 \/ \/ \/ 1218375000 676875 |
|- -- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------|*|-- - -------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
\ 10 2 2 / \2 2 2 / Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
$$x_{1} = - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}$$
Maxima of the function at points:
$$x_{1} = - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}$$
Decreasing at intervals
$$\left(-\infty, - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}\right] \cup \left[- \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}, \infty\right)$$
Increasing at intervals
$$\left[- \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} - \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}, - \frac{\sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}}{2} + \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}} - \frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{48}{95 \sqrt{\frac{13845841}{11250 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}} + \frac{3721}{75} + 2 \sqrt[3]{\frac{\sqrt{55796572722963}}{676875} + \frac{18598860004321}{1218375000}}}} + \frac{7442}{75}}}{2} + \frac{69}{10}\right]$$
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{24 \left(5 - \frac{5 \left(10 x - 69\right)}{5 x - 4} - \frac{10 x - 69}{x - 13}\right)}{\left(x - 13\right)^{2} \left(5 x - 4\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{24 \left(5 - \frac{5 \left(10 x - 69\right)}{5 x - 4} - \frac{10 x - 69}{x - 13}\right)}{\left(x - 13\right)^{2} \left(5 x - 4\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 0.8$$
$$x_{2} = 13$$
$$x_{1} = 0.8$$
$$x_{2} = 13$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 19*x - 12*1/((x - 13)*(5*x - 4)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)}}{x}\right) = 19$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = 19 x$$
$$\lim_{x \to \infty}\left(\frac{19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)}}{x}\right) = 19$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = 19 x$$
$$\lim_{x \to -\infty}\left(\frac{19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)}}{x}\right) = 19$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = 19 x$$
$$\lim_{x \to \infty}\left(\frac{19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)}}{x}\right) = 19$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = 19 x$$
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
$$19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)} = - 19 x - \frac{12}{\left(- 5 x - 4\right) \left(- x - 13\right)}$$
- No
$$19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)} = 19 x + \frac{12}{\left(- 5 x - 4\right) \left(- x - 13\right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)} = - 19 x - \frac{12}{\left(- 5 x - 4\right) \left(- x - 13\right)}$$
- No
$$19 x - \frac{12}{\left(x - 13\right) \left(5 x - 4\right)} = 19 x + \frac{12}{\left(- 5 x - 4\right) \left(- x - 13\right)}$$
- No
so, the function
not is
neither even, nor odd