Mister Exam
Graphing y = arccos(x)-(x)
The solution
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f(x) = acos(x) - x
$$f{\left(x \right)} = - x + \operatorname{acos}{\left(x \right)}$$
f = -x + acos(x)
The graph of the function
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:
$$- x + \operatorname{acos}{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0.739085133215161$$
so we need to solve the equation:
$$- x + \operatorname{acos}{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0.739085133215161$$
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
$$-1 - \frac{1}{\sqrt{1 - x^{2}}} = 0$$
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
$$-1 - \frac{1}{\sqrt{1 - x^{2}}} = 0$$
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
$$- \frac{x}{\left(1 - x^{2}\right)^{\frac{3}{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
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
$$\left(-\infty, 0\right]$$
Convex at the intervals
$$\left[0, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{x}{\left(1 - x^{2}\right)^{\frac{3}{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
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
$$\left(-\infty, 0\right]$$
Convex at the intervals
$$\left[0, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(- x + \operatorname{acos}{\left(x \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- x + \operatorname{acos}{\left(x \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(- x + \operatorname{acos}{\left(x \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- x + \operatorname{acos}{\left(x \right)}\right) = -\infty$$
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 acos(x) - x, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{- x + \operatorname{acos}{\left(x \right)}}{x}\right) = -1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - x$$
$$\lim_{x \to \infty}\left(\frac{- x + \operatorname{acos}{\left(x \right)}}{x}\right) = -1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - x$$
$$\lim_{x \to -\infty}\left(\frac{- x + \operatorname{acos}{\left(x \right)}}{x}\right) = -1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - x$$
$$\lim_{x \to \infty}\left(\frac{- x + \operatorname{acos}{\left(x \right)}}{x}\right) = -1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = - x$$
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:
$$- x + \operatorname{acos}{\left(x \right)} = x + \operatorname{acos}{\left(- x \right)}$$
- No
$$- x + \operatorname{acos}{\left(x \right)} = - x - \operatorname{acos}{\left(- x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- x + \operatorname{acos}{\left(x \right)} = x + \operatorname{acos}{\left(- x \right)}$$
- No
$$- x + \operatorname{acos}{\left(x \right)} = - x - \operatorname{acos}{\left(- x \right)}$$
- No
so, the function
not is
neither even, nor odd