Mister Exam
Graphing y = xarccos(x/(1-x^2))
The solution
You have entered
[src]
/ x \
f(x) = x*acos|------|
| 2|
\1 - x /
$$f{\left(x \right)} = x \operatorname{acos}{\left(\frac{x}{1 - x^{2}} \right)}$$
f = x*acos(x/(1 - x^2))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$x_{1} = -1$$
$$x_{2} = 1$$
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:
$$x \operatorname{acos}{\left(\frac{x}{1 - x^{2}} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = - \frac{1}{2} + \frac{\sqrt{5}}{2}$$
$$x_{3} = - \frac{\sqrt{5}}{2} - \frac{1}{2}$$
Numerical solution
$$x_{1} = 0$$
$$x_{2} = 0.618033988749895$$
$$x_{3} = -1.61803398874989$$
so we need to solve the equation:
$$x \operatorname{acos}{\left(\frac{x}{1 - x^{2}} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = - \frac{1}{2} + \frac{\sqrt{5}}{2}$$
$$x_{3} = - \frac{\sqrt{5}}{2} - \frac{1}{2}$$
Numerical solution
$$x_{1} = 0$$
$$x_{2} = 0.618033988749895$$
$$x_{3} = -1.61803398874989$$
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
$$- \frac{\frac{x^{2} \left(- \frac{8 x^{2}}{x^{2} - 1} + 6 + \frac{\left(\frac{2 x^{2}}{x^{2} - 1} - 1\right)^{2}}{\left(x^{2} - 1\right) \left(\frac{x^{2}}{\left(x^{2} - 1\right)^{2}} - 1\right)}\right)}{x^{2} - 1} + \frac{4 x^{2}}{x^{2} - 1} - 2}{\left(x^{2} - 1\right) \sqrt{- \frac{x^{2}}{\left(1 - x^{2}\right)^{2}} + 1}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -1.35560118154383$$
$$x_{2} = -0.69551980859689$$
$$x_{3} = 0.69551980859689$$
$$x_{4} = 1.35560118154383$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$\lim_{x \to -1^-}\left(- \frac{\frac{x^{2} \left(- \frac{8 x^{2}}{x^{2} - 1} + 6 + \frac{\left(\frac{2 x^{2}}{x^{2} - 1} - 1\right)^{2}}{\left(x^{2} - 1\right) \left(\frac{x^{2}}{\left(x^{2} - 1\right)^{2}} - 1\right)}\right)}{x^{2} - 1} + \frac{4 x^{2}}{x^{2} - 1} - 2}{\left(x^{2} - 1\right) \sqrt{- \frac{x^{2}}{\left(1 - x^{2}\right)^{2}} + 1}}\right) = - \infty i$$
$$\lim_{x \to -1^+}\left(- \frac{\frac{x^{2} \left(- \frac{8 x^{2}}{x^{2} - 1} + 6 + \frac{\left(\frac{2 x^{2}}{x^{2} - 1} - 1\right)^{2}}{\left(x^{2} - 1\right) \left(\frac{x^{2}}{\left(x^{2} - 1\right)^{2}} - 1\right)}\right)}{x^{2} - 1} + \frac{4 x^{2}}{x^{2} - 1} - 2}{\left(x^{2} - 1\right) \sqrt{- \frac{x^{2}}{\left(1 - x^{2}\right)^{2}} + 1}}\right) = \infty i$$
- the limits are not equal, so
$$x_{1} = -1$$
- is an inflection point
$$\lim_{x \to 1^-}\left(- \frac{\frac{x^{2} \left(- \frac{8 x^{2}}{x^{2} - 1} + 6 + \frac{\left(\frac{2 x^{2}}{x^{2} - 1} - 1\right)^{2}}{\left(x^{2} - 1\right) \left(\frac{x^{2}}{\left(x^{2} - 1\right)^{2}} - 1\right)}\right)}{x^{2} - 1} + \frac{4 x^{2}}{x^{2} - 1} - 2}{\left(x^{2} - 1\right) \sqrt{- \frac{x^{2}}{\left(1 - x^{2}\right)^{2}} + 1}}\right) = \infty i$$
$$\lim_{x \to 1^+}\left(- \frac{\frac{x^{2} \left(- \frac{8 x^{2}}{x^{2} - 1} + 6 + \frac{\left(\frac{2 x^{2}}{x^{2} - 1} - 1\right)^{2}}{\left(x^{2} - 1\right) \left(\frac{x^{2}}{\left(x^{2} - 1\right)^{2}} - 1\right)}\right)}{x^{2} - 1} + \frac{4 x^{2}}{x^{2} - 1} - 2}{\left(x^{2} - 1\right) \sqrt{- \frac{x^{2}}{\left(1 - x^{2}\right)^{2}} + 1}}\right) = - \infty i$$
- the limits are not equal, so
$$x_{2} = 1$$
- 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:
Have no bends at the whole real axis
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{\frac{x^{2} \left(- \frac{8 x^{2}}{x^{2} - 1} + 6 + \frac{\left(\frac{2 x^{2}}{x^{2} - 1} - 1\right)^{2}}{\left(x^{2} - 1\right) \left(\frac{x^{2}}{\left(x^{2} - 1\right)^{2}} - 1\right)}\right)}{x^{2} - 1} + \frac{4 x^{2}}{x^{2} - 1} - 2}{\left(x^{2} - 1\right) \sqrt{- \frac{x^{2}}{\left(1 - x^{2}\right)^{2}} + 1}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -1.35560118154383$$
$$x_{2} = -0.69551980859689$$
$$x_{3} = 0.69551980859689$$
$$x_{4} = 1.35560118154383$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$\lim_{x \to -1^-}\left(- \frac{\frac{x^{2} \left(- \frac{8 x^{2}}{x^{2} - 1} + 6 + \frac{\left(\frac{2 x^{2}}{x^{2} - 1} - 1\right)^{2}}{\left(x^{2} - 1\right) \left(\frac{x^{2}}{\left(x^{2} - 1\right)^{2}} - 1\right)}\right)}{x^{2} - 1} + \frac{4 x^{2}}{x^{2} - 1} - 2}{\left(x^{2} - 1\right) \sqrt{- \frac{x^{2}}{\left(1 - x^{2}\right)^{2}} + 1}}\right) = - \infty i$$
$$\lim_{x \to -1^+}\left(- \frac{\frac{x^{2} \left(- \frac{8 x^{2}}{x^{2} - 1} + 6 + \frac{\left(\frac{2 x^{2}}{x^{2} - 1} - 1\right)^{2}}{\left(x^{2} - 1\right) \left(\frac{x^{2}}{\left(x^{2} - 1\right)^{2}} - 1\right)}\right)}{x^{2} - 1} + \frac{4 x^{2}}{x^{2} - 1} - 2}{\left(x^{2} - 1\right) \sqrt{- \frac{x^{2}}{\left(1 - x^{2}\right)^{2}} + 1}}\right) = \infty i$$
- the limits are not equal, so
$$x_{1} = -1$$
- is an inflection point
$$\lim_{x \to 1^-}\left(- \frac{\frac{x^{2} \left(- \frac{8 x^{2}}{x^{2} - 1} + 6 + \frac{\left(\frac{2 x^{2}}{x^{2} - 1} - 1\right)^{2}}{\left(x^{2} - 1\right) \left(\frac{x^{2}}{\left(x^{2} - 1\right)^{2}} - 1\right)}\right)}{x^{2} - 1} + \frac{4 x^{2}}{x^{2} - 1} - 2}{\left(x^{2} - 1\right) \sqrt{- \frac{x^{2}}{\left(1 - x^{2}\right)^{2}} + 1}}\right) = \infty i$$
$$\lim_{x \to 1^+}\left(- \frac{\frac{x^{2} \left(- \frac{8 x^{2}}{x^{2} - 1} + 6 + \frac{\left(\frac{2 x^{2}}{x^{2} - 1} - 1\right)^{2}}{\left(x^{2} - 1\right) \left(\frac{x^{2}}{\left(x^{2} - 1\right)^{2}} - 1\right)}\right)}{x^{2} - 1} + \frac{4 x^{2}}{x^{2} - 1} - 2}{\left(x^{2} - 1\right) \sqrt{- \frac{x^{2}}{\left(1 - x^{2}\right)^{2}} + 1}}\right) = - \infty i$$
- the limits are not equal, so
$$x_{2} = 1$$
- 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:
Have no bends at the whole real axis
Vertical asymptotes
Have:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$x_{1} = -1$$
$$x_{2} = 1$$
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:
$$x \operatorname{acos}{\left(\frac{x}{1 - x^{2}} \right)} = - x \operatorname{acos}{\left(- \frac{x}{1 - x^{2}} \right)}$$
- No
$$x \operatorname{acos}{\left(\frac{x}{1 - x^{2}} \right)} = x \operatorname{acos}{\left(- \frac{x}{1 - x^{2}} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$x \operatorname{acos}{\left(\frac{x}{1 - x^{2}} \right)} = - x \operatorname{acos}{\left(- \frac{x}{1 - x^{2}} \right)}$$
- No
$$x \operatorname{acos}{\left(\frac{x}{1 - x^{2}} \right)} = x \operatorname{acos}{\left(- \frac{x}{1 - x^{2}} \right)}$$
- No
so, the function
not is
neither even, nor odd