Mister Exam
Graphing y = 2ctg(x/2-pi/6)
The solution
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/x pi\
f(x) = 2*cot|- - --|
\2 6 /
$$f{\left(x \right)} = 2 \cot{\left(\frac{x}{2} - \frac{\pi}{6} \right)}$$
f = 2*cot(x/2 - pi/6)
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:
$$2 \cot{\left(\frac{x}{2} - \frac{\pi}{6} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{2 \pi}{3}$$
Numerical solution
$$x_{1} = 60.7374579694027$$
$$x_{2} = 4.18879020478639$$
$$x_{3} = -14.6607657167524$$
$$x_{4} = 92.1533845053006$$
$$x_{5} = 29.3215314335047$$
$$x_{6} = -102.625360017267$$
$$x_{7} = -2.0943951023932$$
$$x_{8} = -83.7758040957278$$
$$x_{9} = -71.2094334813686$$
$$x_{10} = -96.342174710087$$
$$x_{11} = 41.8879020478639$$
$$x_{12} = -33.5103216382911$$
$$x_{13} = -27.2271363311115$$
$$x_{14} = -77.4926187885482$$
$$x_{15} = -64.9262481741891$$
$$x_{16} = -90.0589894029074$$
$$x_{17} = 67.0206432765823$$
$$x_{18} = 48.1710873550435$$
$$x_{19} = 85.870199198121$$
$$x_{20} = -46.0766922526503$$
$$x_{21} = 23.0383461263252$$
$$x_{22} = 16.7551608191456$$
$$x_{23} = -39.7935069454707$$
$$x_{24} = -52.3598775598299$$
$$x_{25} = -58.6430628670095$$
$$x_{26} = 10.471975511966$$
$$x_{27} = 98.4365698124802$$
$$x_{28} = -8.37758040957278$$
$$x_{29} = 79.5870138909414$$
$$x_{30} = 73.3038285837618$$
$$x_{31} = 35.6047167406843$$
$$x_{32} = 54.4542726622231$$
$$x_{33} = -20.943951023932$$
so we need to solve the equation:
$$2 \cot{\left(\frac{x}{2} - \frac{\pi}{6} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{2 \pi}{3}$$
Numerical solution
$$x_{1} = 60.7374579694027$$
$$x_{2} = 4.18879020478639$$
$$x_{3} = -14.6607657167524$$
$$x_{4} = 92.1533845053006$$
$$x_{5} = 29.3215314335047$$
$$x_{6} = -102.625360017267$$
$$x_{7} = -2.0943951023932$$
$$x_{8} = -83.7758040957278$$
$$x_{9} = -71.2094334813686$$
$$x_{10} = -96.342174710087$$
$$x_{11} = 41.8879020478639$$
$$x_{12} = -33.5103216382911$$
$$x_{13} = -27.2271363311115$$
$$x_{14} = -77.4926187885482$$
$$x_{15} = -64.9262481741891$$
$$x_{16} = -90.0589894029074$$
$$x_{17} = 67.0206432765823$$
$$x_{18} = 48.1710873550435$$
$$x_{19} = 85.870199198121$$
$$x_{20} = -46.0766922526503$$
$$x_{21} = 23.0383461263252$$
$$x_{22} = 16.7551608191456$$
$$x_{23} = -39.7935069454707$$
$$x_{24} = -52.3598775598299$$
$$x_{25} = -58.6430628670095$$
$$x_{26} = 10.471975511966$$
$$x_{27} = 98.4365698124802$$
$$x_{28} = -8.37758040957278$$
$$x_{29} = 79.5870138909414$$
$$x_{30} = 73.3038285837618$$
$$x_{31} = 35.6047167406843$$
$$x_{32} = 54.4542726622231$$
$$x_{33} = -20.943951023932$$
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
$$- \cot^{2}{\left(\frac{x}{2} - \frac{\pi}{6} \right)} - 1 = 0$$
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
$$- \cot^{2}{\left(\frac{x}{2} - \frac{\pi}{6} \right)} - 1 = 0$$
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
$$\left(\cot^{2}{\left(\frac{3 x - \pi}{6} \right)} + 1\right) \cot{\left(\frac{3 x - \pi}{6} \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{2 \pi}{3}$$
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
$$\left(-\infty, - \frac{2 \pi}{3}\right]$$
Convex at the intervals
$$\left[- \frac{2 \pi}{3}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\left(\cot^{2}{\left(\frac{3 x - \pi}{6} \right)} + 1\right) \cot{\left(\frac{3 x - \pi}{6} \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{2 \pi}{3}$$
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
$$\left(-\infty, - \frac{2 \pi}{3}\right]$$
Convex at the intervals
$$\left[- \frac{2 \pi}{3}, \infty\right)$$
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:
$$y = \lim_{x \to -\infty}\left(2 \cot{\left(\frac{x}{2} - \frac{\pi}{6} \right)}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(2 \cot{\left(\frac{x}{2} - \frac{\pi}{6} \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(2 \cot{\left(\frac{x}{2} - \frac{\pi}{6} \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(2 \cot{\left(\frac{x}{2} - \frac{\pi}{6} \right)}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*cot(x/2 - pi/6), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{2 \cot{\left(\frac{x}{2} - \frac{\pi}{6} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{2 \cot{\left(\frac{x}{2} - \frac{\pi}{6} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{2 \cot{\left(\frac{x}{2} - \frac{\pi}{6} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{2 \cot{\left(\frac{x}{2} - \frac{\pi}{6} \right)}}{x}\right)$$
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:
$$2 \cot{\left(\frac{x}{2} - \frac{\pi}{6} \right)} = - 2 \cot{\left(\frac{x}{2} + \frac{\pi}{6} \right)}$$
- No
$$2 \cot{\left(\frac{x}{2} - \frac{\pi}{6} \right)} = 2 \cot{\left(\frac{x}{2} + \frac{\pi}{6} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$2 \cot{\left(\frac{x}{2} - \frac{\pi}{6} \right)} = - 2 \cot{\left(\frac{x}{2} + \frac{\pi}{6} \right)}$$
- No
$$2 \cot{\left(\frac{x}{2} - \frac{\pi}{6} \right)} = 2 \cot{\left(\frac{x}{2} + \frac{\pi}{6} \right)}$$
- No
so, the function
not is
neither even, nor odd