Mister Exam
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-
How to use it?
- Graphing y =:
- x^3-27x
- x^2+4x-3
- x^2/(4-x^2)
- -x^2-4x+12
-
Identical expressions
- (2lnx/x((x+ three)/x))- three
- (2lnx divide by x((x plus 3) divide by x)) minus 3
- (2lnx divide by x((x plus three) divide by x)) minus three
- 2lnx/xx+3/x-3
- (2lnx divide by x((x+3) divide by x))-3
-
Similar expressions
- (2lnx/x((x+3)/x))+3
- (2lnx/x((x-3)/x))-3
Graphing y = (2lnx/x((x+3)/x))-3
The solution
You have entered
[src]
2*log(x)*(x + 3)
f(x) = ---------------- - 3
x*x
$$f{\left(x \right)} = \frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3$$
f = -1*3 + 2*(x + 3)*log(x)/(x*x)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3 = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3 = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$\frac{2 \left(- 4 \log{\left(x \right)} + \frac{6 \left(x + 3\right) \log{\left(x \right)}}{x} + 2 - \frac{5 \left(x + 3\right)}{x}\right)}{x^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 43847.5905884662$$
$$x_{2} = 53841.952602536$$
$$x_{3} = 31502.6420619197$$
$$x_{4} = 34886.2640297897$$
$$x_{5} = 52735.3601050521$$
$$x_{6} = 33759.938295641$$
$$x_{7} = 2.68147262116116$$
$$x_{8} = 32632.0827716515$$
$$x_{9} = 57156.4807890037$$
$$x_{10} = 30371.5579010177$$
$$x_{11} = 48299.6168667142$$
$$x_{12} = 36011.1126055766$$
$$x_{13} = 40496.6981287632$$
$$x_{14} = 49410.0091923193$$
$$x_{15} = 44962.2264158991$$
$$x_{16} = 50519.4130981396$$
$$x_{17} = 54947.6556019799$$
$$x_{18} = 47188.2084874369$$
$$x_{19} = 37134.5340105345$$
$$x_{20} = 46075.7551136742$$
$$x_{21} = 51627.8549875932$$
$$x_{22} = 41614.8623757666$$
$$x_{23} = 39377.2828114816$$
$$x_{24} = 38256.5757166382$$
$$x_{25} = 42731.8142563409$$
$$x_{26} = 56052.4912533661$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(- 4 \log{\left(x \right)} + \frac{6 \left(x + 3\right) \log{\left(x \right)}}{x} + 2 - \frac{5 \left(x + 3\right)}{x}\right)}{x^{3}}\right) = -\infty$$
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{2 \left(- 4 \log{\left(x \right)} + \frac{6 \left(x + 3\right) \log{\left(x \right)}}{x} + 2 - \frac{5 \left(x + 3\right)}{x}\right)}{x^{3}}\right) = -\infty$$
Let's take the limit
- limits are equal, then skip the corresponding point
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[2.68147262116116, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 2.68147262116116\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2 \left(- 4 \log{\left(x \right)} + \frac{6 \left(x + 3\right) \log{\left(x \right)}}{x} + 2 - \frac{5 \left(x + 3\right)}{x}\right)}{x^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 43847.5905884662$$
$$x_{2} = 53841.952602536$$
$$x_{3} = 31502.6420619197$$
$$x_{4} = 34886.2640297897$$
$$x_{5} = 52735.3601050521$$
$$x_{6} = 33759.938295641$$
$$x_{7} = 2.68147262116116$$
$$x_{8} = 32632.0827716515$$
$$x_{9} = 57156.4807890037$$
$$x_{10} = 30371.5579010177$$
$$x_{11} = 48299.6168667142$$
$$x_{12} = 36011.1126055766$$
$$x_{13} = 40496.6981287632$$
$$x_{14} = 49410.0091923193$$
$$x_{15} = 44962.2264158991$$
$$x_{16} = 50519.4130981396$$
$$x_{17} = 54947.6556019799$$
$$x_{18} = 47188.2084874369$$
$$x_{19} = 37134.5340105345$$
$$x_{20} = 46075.7551136742$$
$$x_{21} = 51627.8549875932$$
$$x_{22} = 41614.8623757666$$
$$x_{23} = 39377.2828114816$$
$$x_{24} = 38256.5757166382$$
$$x_{25} = 42731.8142563409$$
$$x_{26} = 56052.4912533661$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(- 4 \log{\left(x \right)} + \frac{6 \left(x + 3\right) \log{\left(x \right)}}{x} + 2 - \frac{5 \left(x + 3\right)}{x}\right)}{x^{3}}\right) = -\infty$$
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{2 \left(- 4 \log{\left(x \right)} + \frac{6 \left(x + 3\right) \log{\left(x \right)}}{x} + 2 - \frac{5 \left(x + 3\right)}{x}\right)}{x^{3}}\right) = -\infty$$
Let's take the limit
- limits are equal, then skip the corresponding point
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[2.68147262116116, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 2.68147262116116\right]$$
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3\right) = -3$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -3$$
$$\lim_{x \to \infty}\left(\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3\right) = -3$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -3$$
$$\lim_{x \to -\infty}\left(\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3\right) = -3$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -3$$
$$\lim_{x \to \infty}\left(\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3\right) = -3$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -3$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*log(x)*(x + 3)/(x*x) - 1*3, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
$$\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3 = -3 + \frac{2 \cdot \left(- x + 3\right) \log{\left(- x \right)}}{x^{2}}$$
- No
$$\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3 = 3 - \frac{2 \cdot \left(- x + 3\right) \log{\left(- x \right)}}{x^{2}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3 = -3 + \frac{2 \cdot \left(- x + 3\right) \log{\left(- x \right)}}{x^{2}}$$
- No
$$\frac{2 \left(x + 3\right) \log{\left(x \right)}}{x x} - 3 = 3 - \frac{2 \cdot \left(- x + 3\right) \log{\left(- x \right)}}{x^{2}}$$
- No
so, the function
not is
neither even, nor odd
The graph