Mister Exam
Graphing y = -(2-x)/((4*x+5-x^2)*sqrt(4*x+5-x^2))
The solution
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-2 + x
f(x) = --------------------------------
______________
/ 2\ / 2
\4*x + 5 - x /*\/ 4*x + 5 - x
$$f{\left(x \right)} = \frac{x - 2}{\sqrt{- x^{2} + \left(4 x + 5\right)} \left(- x^{2} + \left(4 x + 5\right)\right)}$$
f = (x - 2)/((sqrt(-x^2 + 4*x + 5)*(-x^2 + 4*x + 5)))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -1$$
$$x_{2} = 5$$
$$x_{1} = -1$$
$$x_{2} = 5$$
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:
$$\frac{x - 2}{\sqrt{- x^{2} + \left(4 x + 5\right)} \left(- x^{2} + \left(4 x + 5\right)\right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 2$$
Numerical solution
$$x_{1} = 2$$
so we need to solve the equation:
$$\frac{x - 2}{\sqrt{- x^{2} + \left(4 x + 5\right)} \left(- x^{2} + \left(4 x + 5\right)\right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 2$$
Numerical solution
$$x_{1} = 2$$
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
$$\frac{\left(x - 2\right) \left(- \left(2 - x\right) \sqrt{- x^{2} + \left(4 x + 5\right)} - \left(4 - 2 x\right) \sqrt{- x^{2} + \left(4 x + 5\right)}\right)}{\left(- x^{2} + \left(4 x + 5\right)\right)^{3}} + \frac{1}{\sqrt{- x^{2} + \left(4 x + 5\right)} \left(- x^{2} + \left(4 x + 5\right)\right)} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{\left(x - 2\right) \left(- \left(2 - x\right) \sqrt{- x^{2} + \left(4 x + 5\right)} - \left(4 - 2 x\right) \sqrt{- x^{2} + \left(4 x + 5\right)}\right)}{\left(- x^{2} + \left(4 x + 5\right)\right)^{3}} + \frac{1}{\sqrt{- x^{2} + \left(4 x + 5\right)} \left(- x^{2} + \left(4 x + 5\right)\right)} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
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
$$3 \left(x - 2\right) \left(\frac{6 \left(x - 2\right)^{2}}{\left(- x^{2} + 4 x + 5\right)^{\frac{7}{2}}} - \frac{\frac{\left(x - 2\right)^{2}}{\sqrt{- x^{2} + 4 x + 5}} - \sqrt{- x^{2} + 4 x + 5}}{\left(- x^{2} + 4 x + 5\right)^{3}} + \frac{2}{\left(- x^{2} + 4 x + 5\right)^{\frac{5}{2}}}\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 2$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -1$$
$$x_{2} = 5$$
$$\lim_{x \to -1^-}\left(3 \left(x - 2\right) \left(\frac{6 \left(x - 2\right)^{2}}{\left(- x^{2} + 4 x + 5\right)^{\frac{7}{2}}} - \frac{\frac{\left(x - 2\right)^{2}}{\sqrt{- x^{2} + 4 x + 5}} - \sqrt{- x^{2} + 4 x + 5}}{\left(- x^{2} + 4 x + 5\right)^{3}} + \frac{2}{\left(- x^{2} + 4 x + 5\right)^{\frac{5}{2}}}\right)\right) = - \infty i$$
$$\lim_{x \to -1^+}\left(3 \left(x - 2\right) \left(\frac{6 \left(x - 2\right)^{2}}{\left(- x^{2} + 4 x + 5\right)^{\frac{7}{2}}} - \frac{\frac{\left(x - 2\right)^{2}}{\sqrt{- x^{2} + 4 x + 5}} - \sqrt{- x^{2} + 4 x + 5}}{\left(- x^{2} + 4 x + 5\right)^{3}} + \frac{2}{\left(- x^{2} + 4 x + 5\right)^{\frac{5}{2}}}\right)\right) = -\infty$$
- the limits are not equal, so
$$x_{1} = -1$$
- is an inflection point
$$\lim_{x \to 5^-}\left(3 \left(x - 2\right) \left(\frac{6 \left(x - 2\right)^{2}}{\left(- x^{2} + 4 x + 5\right)^{\frac{7}{2}}} - \frac{\frac{\left(x - 2\right)^{2}}{\sqrt{- x^{2} + 4 x + 5}} - \sqrt{- x^{2} + 4 x + 5}}{\left(- x^{2} + 4 x + 5\right)^{3}} + \frac{2}{\left(- x^{2} + 4 x + 5\right)^{\frac{5}{2}}}\right)\right) = \infty$$
$$\lim_{x \to 5^+}\left(3 \left(x - 2\right) \left(\frac{6 \left(x - 2\right)^{2}}{\left(- x^{2} + 4 x + 5\right)^{\frac{7}{2}}} - \frac{\frac{\left(x - 2\right)^{2}}{\sqrt{- x^{2} + 4 x + 5}} - \sqrt{- x^{2} + 4 x + 5}}{\left(- x^{2} + 4 x + 5\right)^{3}} + \frac{2}{\left(- x^{2} + 4 x + 5\right)^{\frac{5}{2}}}\right)\right) = \infty i$$
- the limits are not equal, so
$$x_{2} = 5$$
- is an inflection 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
$$\left[2, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 2\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$3 \left(x - 2\right) \left(\frac{6 \left(x - 2\right)^{2}}{\left(- x^{2} + 4 x + 5\right)^{\frac{7}{2}}} - \frac{\frac{\left(x - 2\right)^{2}}{\sqrt{- x^{2} + 4 x + 5}} - \sqrt{- x^{2} + 4 x + 5}}{\left(- x^{2} + 4 x + 5\right)^{3}} + \frac{2}{\left(- x^{2} + 4 x + 5\right)^{\frac{5}{2}}}\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 2$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -1$$
$$x_{2} = 5$$
$$\lim_{x \to -1^-}\left(3 \left(x - 2\right) \left(\frac{6 \left(x - 2\right)^{2}}{\left(- x^{2} + 4 x + 5\right)^{\frac{7}{2}}} - \frac{\frac{\left(x - 2\right)^{2}}{\sqrt{- x^{2} + 4 x + 5}} - \sqrt{- x^{2} + 4 x + 5}}{\left(- x^{2} + 4 x + 5\right)^{3}} + \frac{2}{\left(- x^{2} + 4 x + 5\right)^{\frac{5}{2}}}\right)\right) = - \infty i$$
$$\lim_{x \to -1^+}\left(3 \left(x - 2\right) \left(\frac{6 \left(x - 2\right)^{2}}{\left(- x^{2} + 4 x + 5\right)^{\frac{7}{2}}} - \frac{\frac{\left(x - 2\right)^{2}}{\sqrt{- x^{2} + 4 x + 5}} - \sqrt{- x^{2} + 4 x + 5}}{\left(- x^{2} + 4 x + 5\right)^{3}} + \frac{2}{\left(- x^{2} + 4 x + 5\right)^{\frac{5}{2}}}\right)\right) = -\infty$$
- the limits are not equal, so
$$x_{1} = -1$$
- is an inflection point
$$\lim_{x \to 5^-}\left(3 \left(x - 2\right) \left(\frac{6 \left(x - 2\right)^{2}}{\left(- x^{2} + 4 x + 5\right)^{\frac{7}{2}}} - \frac{\frac{\left(x - 2\right)^{2}}{\sqrt{- x^{2} + 4 x + 5}} - \sqrt{- x^{2} + 4 x + 5}}{\left(- x^{2} + 4 x + 5\right)^{3}} + \frac{2}{\left(- x^{2} + 4 x + 5\right)^{\frac{5}{2}}}\right)\right) = \infty$$
$$\lim_{x \to 5^+}\left(3 \left(x - 2\right) \left(\frac{6 \left(x - 2\right)^{2}}{\left(- x^{2} + 4 x + 5\right)^{\frac{7}{2}}} - \frac{\frac{\left(x - 2\right)^{2}}{\sqrt{- x^{2} + 4 x + 5}} - \sqrt{- x^{2} + 4 x + 5}}{\left(- x^{2} + 4 x + 5\right)^{3}} + \frac{2}{\left(- x^{2} + 4 x + 5\right)^{\frac{5}{2}}}\right)\right) = \infty i$$
- the limits are not equal, so
$$x_{2} = 5$$
- is an inflection 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
$$\left[2, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 2\right]$$
Vertical asymptotes
Have:
$$x_{1} = -1$$
$$x_{2} = 5$$
$$x_{1} = -1$$
$$x_{2} = 5$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x - 2}{\sqrt{- x^{2} + \left(4 x + 5\right)} \left(- x^{2} + \left(4 x + 5\right)\right)}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{x - 2}{\sqrt{- x^{2} + \left(4 x + 5\right)} \left(- x^{2} + \left(4 x + 5\right)\right)}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{x - 2}{\sqrt{- x^{2} + \left(4 x + 5\right)} \left(- x^{2} + \left(4 x + 5\right)\right)}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{x - 2}{\sqrt{- x^{2} + \left(4 x + 5\right)} \left(- x^{2} + \left(4 x + 5\right)\right)}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (-2 + x)/(((4*x + 5 - x^2)*sqrt(4*x + 5 - x^2))), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{x - 2}{x \left(- x^{2} + \left(4 x + 5\right)\right)^{\frac{3}{2}}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x - 2}{x \left(- x^{2} + \left(4 x + 5\right)\right)^{\frac{3}{2}}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{x - 2}{x \left(- x^{2} + \left(4 x + 5\right)\right)^{\frac{3}{2}}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{x - 2}{x \left(- x^{2} + \left(4 x + 5\right)\right)^{\frac{3}{2}}}\right) = 0$$
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:
$$\frac{x - 2}{\sqrt{- x^{2} + \left(4 x + 5\right)} \left(- x^{2} + \left(4 x + 5\right)\right)} = \frac{- x - 2}{\left(- x^{2} - 4 x + 5\right)^{\frac{3}{2}}}$$
- No
$$\frac{x - 2}{\sqrt{- x^{2} + \left(4 x + 5\right)} \left(- x^{2} + \left(4 x + 5\right)\right)} = - \frac{- x - 2}{\left(- x^{2} - 4 x + 5\right)^{\frac{3}{2}}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{x - 2}{\sqrt{- x^{2} + \left(4 x + 5\right)} \left(- x^{2} + \left(4 x + 5\right)\right)} = \frac{- x - 2}{\left(- x^{2} - 4 x + 5\right)^{\frac{3}{2}}}$$
- No
$$\frac{x - 2}{\sqrt{- x^{2} + \left(4 x + 5\right)} \left(- x^{2} + \left(4 x + 5\right)\right)} = - \frac{- x - 2}{\left(- x^{2} - 4 x + 5\right)^{\frac{3}{2}}}$$
- No
so, the function
not is
neither even, nor odd