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Plotting a function graph/ 2ctg(x/2-pi/6)

Graphing y = 2ctg(x/2-pi/6)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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            /x   pi\
f(x) = 2*cot|- - --|
            \2   6 /
f(x)=2cot⁡(x2−π6)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
7.00.01.02.03.04.05.06.0-1.0-2000020000
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:
2cot⁡(x2−π6)=02 \cot{\left(\frac{x}{2} - \frac{\pi}{6} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−2π3x_{1} = - \frac{2 \pi}{3}
Numerical solution
x1=60.7374579694027x_{1} = 60.7374579694027
x2=4.18879020478639x_{2} = 4.18879020478639
x3=−14.6607657167524x_{3} = -14.6607657167524
x4=92.1533845053006x_{4} = 92.1533845053006
x5=29.3215314335047x_{5} = 29.3215314335047
x6=−102.625360017267x_{6} = -102.625360017267
x7=−2.0943951023932x_{7} = -2.0943951023932
x8=−83.7758040957278x_{8} = -83.7758040957278
x9=−71.2094334813686x_{9} = -71.2094334813686
x10=−96.342174710087x_{10} = -96.342174710087
x11=41.8879020478639x_{11} = 41.8879020478639
x12=−33.5103216382911x_{12} = -33.5103216382911
x13=−27.2271363311115x_{13} = -27.2271363311115
x14=−77.4926187885482x_{14} = -77.4926187885482
x15=−64.9262481741891x_{15} = -64.9262481741891
x16=−90.0589894029074x_{16} = -90.0589894029074
x17=67.0206432765823x_{17} = 67.0206432765823
x18=48.1710873550435x_{18} = 48.1710873550435
x19=85.870199198121x_{19} = 85.870199198121
x20=−46.0766922526503x_{20} = -46.0766922526503
x21=23.0383461263252x_{21} = 23.0383461263252
x22=16.7551608191456x_{22} = 16.7551608191456
x23=−39.7935069454707x_{23} = -39.7935069454707
x24=−52.3598775598299x_{24} = -52.3598775598299
x25=−58.6430628670095x_{25} = -58.6430628670095
x26=10.471975511966x_{26} = 10.471975511966
x27=98.4365698124802x_{27} = 98.4365698124802
x28=−8.37758040957278x_{28} = -8.37758040957278
x29=79.5870138909414x_{29} = 79.5870138909414
x30=73.3038285837618x_{30} = 73.3038285837618
x31=35.6047167406843x_{31} = 35.6047167406843
x32=54.4542726622231x_{32} = 54.4542726622231
x33=−20.943951023932x_{33} = -20.943951023932
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*cot(x/2 - pi/6).
2cot⁡(−π6+02)2 \cot{\left(- \frac{\pi}{6} + \frac{0}{2} \right)}
The result:
f(0)=−23f{\left(0 \right)} = - 2 \sqrt{3}
The point:
(0, -2*sqrt(3))
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
−cot⁡2(x2−π6)−1=0- \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
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
(cot⁡2(3x−π6)+1)cot⁡(3x−π6)=0\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
x1=−2π3x_{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
(−∞,−2π3]\left(-\infty, - \frac{2 \pi}{3}\right]
Convex at the intervals
[−2π3,∞)\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
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞(2cot⁡(x2−π6))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→∞(2cot⁡(x2−π6))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
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(2cot⁡(x2−π6)x)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=xlim⁡x→∞(2cot⁡(x2−π6)x)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:
2cot⁡(x2−π6)=−2cot⁡(x2+π6)2 \cot{\left(\frac{x}{2} - \frac{\pi}{6} \right)} = - 2 \cot{\left(\frac{x}{2} + \frac{\pi}{6} \right)}
- No
2cot⁡(x2−π6)=2cot⁡(x2+π6)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
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