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ctg(x/3)
ln^2x
xe^-3x
log(1/8)*x
ctg(x/3)
Graphing y = ctg(x/3)
The solution
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/x\
f(x) = cot|-|
\3/
$$f{\left(x \right)} = \cot{\left(\frac{x}{3} \right)}$$
f = cot(x/3)
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:
$$\cot{\left(\frac{x}{3} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{3 \pi}{2}$$
Numerical solution
$$x_{1} = 23.5619449019235$$
$$x_{2} = -4.71238898038469$$
$$x_{3} = -23.5619449019235$$
$$x_{4} = 98.9601685880785$$
$$x_{5} = 61.261056745001$$
$$x_{6} = 4.71238898038469$$
$$x_{7} = -51.8362787842316$$
$$x_{8} = -32.9867228626928$$
$$x_{9} = 80.1106126665397$$
$$x_{10} = 51.8362787842316$$
$$x_{11} = -89.5353906273091$$
$$x_{12} = 42.4115008234622$$
$$x_{13} = 89.5353906273091$$
$$x_{14} = -80.1106126665397$$
$$x_{15} = -42.4115008234622$$
$$x_{16} = -98.9601685880785$$
$$x_{17} = -61.261056745001$$
$$x_{18} = -70.6858347057703$$
$$x_{19} = 70.6858347057703$$
$$x_{20} = -14.1371669411541$$
$$x_{21} = 32.9867228626928$$
$$x_{22} = 14.1371669411541$$
so we need to solve the equation:
$$\cot{\left(\frac{x}{3} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{3 \pi}{2}$$
Numerical solution
$$x_{1} = 23.5619449019235$$
$$x_{2} = -4.71238898038469$$
$$x_{3} = -23.5619449019235$$
$$x_{4} = 98.9601685880785$$
$$x_{5} = 61.261056745001$$
$$x_{6} = 4.71238898038469$$
$$x_{7} = -51.8362787842316$$
$$x_{8} = -32.9867228626928$$
$$x_{9} = 80.1106126665397$$
$$x_{10} = 51.8362787842316$$
$$x_{11} = -89.5353906273091$$
$$x_{12} = 42.4115008234622$$
$$x_{13} = 89.5353906273091$$
$$x_{14} = -80.1106126665397$$
$$x_{15} = -42.4115008234622$$
$$x_{16} = -98.9601685880785$$
$$x_{17} = -61.261056745001$$
$$x_{18} = -70.6858347057703$$
$$x_{19} = 70.6858347057703$$
$$x_{20} = -14.1371669411541$$
$$x_{21} = 32.9867228626928$$
$$x_{22} = 14.1371669411541$$
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
$$- \frac{\cot^{2}{\left(\frac{x}{3} \right)}}{3} - \frac{1}{3} = 0$$
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
$$- \frac{\cot^{2}{\left(\frac{x}{3} \right)}}{3} - \frac{1}{3} = 0$$
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
$$\frac{2 \left(\cot^{2}{\left(\frac{x}{3} \right)} + 1\right) \cot{\left(\frac{x}{3} \right)}}{9} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{3 \pi}{2}$$
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
$$\left(-\infty, \frac{3 \pi}{2}\right]$$
Convex at the intervals
$$\left[\frac{3 \pi}{2}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2 \left(\cot^{2}{\left(\frac{x}{3} \right)} + 1\right) \cot{\left(\frac{x}{3} \right)}}{9} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{3 \pi}{2}$$
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
$$\left(-\infty, \frac{3 \pi}{2}\right]$$
Convex at the intervals
$$\left[\frac{3 \pi}{2}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \cot{\left(\frac{x}{3} \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty} \cot{\left(\frac{x}{3} \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty} \cot{\left(\frac{x}{3} \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty} \cot{\left(\frac{x}{3} \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(x/3), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{x}{3} \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{x}{3} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{x}{3} \right)}}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{\cot{\left(\frac{x}{3} \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\cot{\left(\frac{x}{3} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot{\left(\frac{x}{3} \right)}}{x}\right)$$
$$\lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{x}{3} \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{x}{3} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{x}{3} \right)}}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{\cot{\left(\frac{x}{3} \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\cot{\left(\frac{x}{3} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\cot{\left(\frac{x}{3} \right)}}{x}\right)$$
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:
$$\cot{\left(\frac{x}{3} \right)} = - \cot{\left(\frac{x}{3} \right)}$$
- No
$$\cot{\left(\frac{x}{3} \right)} = \cot{\left(\frac{x}{3} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\cot{\left(\frac{x}{3} \right)} = - \cot{\left(\frac{x}{3} \right)}$$
- No
$$\cot{\left(\frac{x}{3} \right)} = \cot{\left(\frac{x}{3} \right)}$$
- No
so, the function
not is
neither even, nor odd
The graph