Mister Exam
Graphing y = (ctg(pix)-1)^(1/3)
The solution
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3 _______________ f(x) = \/ cot(pi*x) - 1
$$f{\left(x \right)} = \sqrt[3]{\cot{\left(\pi x \right)} - 1}$$
f = (cot(pi*x) - 1)^(1/3)
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:
$$\sqrt[3]{\cot{\left(\pi x \right)} - 1} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{4}$$
Numerical solution
$$x_{1} = 0.25$$
so we need to solve the equation:
$$\sqrt[3]{\cot{\left(\pi x \right)} - 1} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{4}$$
Numerical solution
$$x_{1} = 0.25$$
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
$$\frac{\pi \left(- \cot^{2}{\left(\pi x \right)} - 1\right)}{3 \left(\cot{\left(\pi x \right)} - 1\right)^{\frac{2}{3}}} = 0$$
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
$$\frac{\pi \left(- \cot^{2}{\left(\pi x \right)} - 1\right)}{3 \left(\cot{\left(\pi x \right)} - 1\right)^{\frac{2}{3}}} = 0$$
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
$$\frac{2 \pi^{2} \left(3 \cot{\left(\pi x \right)} - \frac{\cot^{2}{\left(\pi x \right)} + 1}{\cot{\left(\pi x \right)} - 1}\right) \left(\cot^{2}{\left(\pi x \right)} + 1\right)}{9 \left(\cot{\left(\pi x \right)} - 1\right)^{\frac{2}{3}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{\operatorname{acot}{\left(\frac{3}{4} - \frac{\sqrt{17}}{4} \right)}}{\pi}$$
$$x_{2} = \frac{\operatorname{acot}{\left(\frac{3}{4} + \frac{\sqrt{17}}{4} \right)}}{\pi}$$
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
$$\left(-\infty, \frac{\operatorname{acot}{\left(\frac{3}{4} + \frac{\sqrt{17}}{4} \right)}}{\pi}\right]$$
Convex at the intervals
$$\left[\frac{\operatorname{acot}{\left(\frac{3}{4} + \frac{\sqrt{17}}{4} \right)}}{\pi}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2 \pi^{2} \left(3 \cot{\left(\pi x \right)} - \frac{\cot^{2}{\left(\pi x \right)} + 1}{\cot{\left(\pi x \right)} - 1}\right) \left(\cot^{2}{\left(\pi x \right)} + 1\right)}{9 \left(\cot{\left(\pi x \right)} - 1\right)^{\frac{2}{3}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{\operatorname{acot}{\left(\frac{3}{4} - \frac{\sqrt{17}}{4} \right)}}{\pi}$$
$$x_{2} = \frac{\operatorname{acot}{\left(\frac{3}{4} + \frac{\sqrt{17}}{4} \right)}}{\pi}$$
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
$$\left(-\infty, \frac{\operatorname{acot}{\left(\frac{3}{4} + \frac{\sqrt{17}}{4} \right)}}{\pi}\right]$$
Convex at the intervals
$$\left[\frac{\operatorname{acot}{\left(\frac{3}{4} + \frac{\sqrt{17}}{4} \right)}}{\pi}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \sqrt[3]{\cot{\left(\pi x \right)} - 1}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \sqrt[3]{\cot{\left(\pi x \right)} - 1}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \sqrt[3]{\cot{\left(\pi x \right)} - 1}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \sqrt[3]{\cot{\left(\pi x \right)} - 1}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (cot(pi*x) - 1)^(1/3), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\sqrt[3]{\cot{\left(\pi x \right)} - 1}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\sqrt[3]{\cot{\left(\pi x \right)} - 1}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\sqrt[3]{\cot{\left(\pi x \right)} - 1}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\sqrt[3]{\cot{\left(\pi x \right)} - 1}}{x}\right)$$
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:
$$\sqrt[3]{\cot{\left(\pi x \right)} - 1} = \sqrt[3]{- \cot{\left(\pi x \right)} - 1}$$
- No
$$\sqrt[3]{\cot{\left(\pi x \right)} - 1} = - \sqrt[3]{- \cot{\left(\pi x \right)} - 1}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sqrt[3]{\cot{\left(\pi x \right)} - 1} = \sqrt[3]{- \cot{\left(\pi x \right)} - 1}$$
- No
$$\sqrt[3]{\cot{\left(\pi x \right)} - 1} = - \sqrt[3]{- \cot{\left(\pi x \right)} - 1}$$
- No
so, the function
not is
neither even, nor odd