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Plotting a function graph/ ctg(3x/5)

Graphing y = ctg(3x/5)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          /3*x\
f(x) = cot|---|
          \ 5 /
f(x)=cot⁡(3x5)f{\left(x \right)} = \cot{\left(\frac{3 x}{5} \right)} 
f = cot((3*x)/5)
The graph of the function
02468-8-6-4-2-1010-100100
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⁡(3x5)=0\cot{\left(\frac{3 x}{5} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=5π6x_{1} = \frac{5 \pi}{6}
Numerical solution
x1=7.85398163397448x_{1} = 7.85398163397448
x2=−54.9778714378214x_{2} = -54.9778714378214
x3=60.2138591938044x_{3} = 60.2138591938044
x4=−96.8657734856853x_{4} = -96.8657734856853
x5=−75.9218224617533x_{5} = -75.9218224617533
x6=102.101761241668x_{6} = 102.101761241668
x7=−44.5058959258554x_{7} = -44.5058959258554
x8=−39.2699081698724x_{8} = -39.2699081698724
x9=2.61799387799149x_{9} = 2.61799387799149
x10=28.7979326579064x_{10} = 28.7979326579064
x11=−34.0339204138894x_{11} = -34.0339204138894
x12=−91.6297857297023x_{12} = -91.6297857297023
x13=54.9778714378214x_{13} = 54.9778714378214
x14=70.6858347057703x_{14} = 70.6858347057703
x15=−49.7418836818384x_{15} = -49.7418836818384
x16=23.5619449019235x_{16} = 23.5619449019235
x17=65.4498469497874x_{17} = 65.4498469497874
x18=75.9218224617533x_{18} = 75.9218224617533
x19=81.1578102177363x_{19} = 81.1578102177363
x20=91.6297857297023x_{20} = 91.6297857297023
x21=−86.3937979737193x_{21} = -86.3937979737193
x22=39.2699081698724x_{22} = 39.2699081698724
x23=−81.1578102177363x_{23} = -81.1578102177363
x24=96.8657734856853x_{24} = 96.8657734856853
x25=−7.85398163397448x_{25} = -7.85398163397448
x26=86.3937979737193x_{26} = 86.3937979737193
x27=−2.61799387799149x_{27} = -2.61799387799149
x28=−13.0899693899575x_{28} = -13.0899693899575
x29=34.0339204138894x_{29} = 34.0339204138894
x30=44.5058959258554x_{30} = 44.5058959258554
x31=−28.7979326579064x_{31} = -28.7979326579064
x32=49.7418836818384x_{32} = 49.7418836818384
x33=13.0899693899575x_{33} = 13.0899693899575
x34=18.3259571459405x_{34} = 18.3259571459405
x35=−18.3259571459405x_{35} = -18.3259571459405
x36=−23.5619449019235x_{36} = -23.5619449019235
x37=−102.101761241668x_{37} = -102.101761241668
x38=−60.2138591938044x_{38} = -60.2138591938044
x39=−70.6858347057703x_{39} = -70.6858347057703
x40=−65.4498469497874x_{40} = -65.4498469497874
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cot((3*x)/5).
cot⁡(0⋅35)\cot{\left(\frac{0 \cdot 3}{5} \right)}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
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
−3cot⁡2(3x5)5−35=0- \frac{3 \cot^{2}{\left(\frac{3 x}{5} \right)}}{5} - \frac{3}{5} = 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
18(cot⁡2(3x5)+1)cot⁡(3x5)25=0\frac{18 \left(\cot^{2}{\left(\frac{3 x}{5} \right)} + 1\right) \cot{\left(\frac{3 x}{5} \right)}}{25} = 0
Solve this equation
The roots of this equation
x1=5π6x_{1} = \frac{5 \pi}{6}

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
(−∞,5π6]\left(-\infty, \frac{5 \pi}{6}\right]
Convex at the intervals
[5π6,∞)\left[\frac{5 \pi}{6}, \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→−∞cot⁡(3x5)=−cot⁡(∞)\lim_{x \to -\infty} \cot{\left(\frac{3 x}{5} \right)} = - \cot{\left(\infty \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=−cot⁡(∞)y = - \cot{\left(\infty \right)}
lim⁡x→∞cot⁡(3x5)=cot⁡(∞)\lim_{x \to \infty} \cot{\left(\frac{3 x}{5} \right)} = \cot{\left(\infty \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=cot⁡(∞)y = \cot{\left(\infty \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot((3*x)/5), 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→−∞(cot⁡(3x5)x)y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{3 x}{5} \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(cot⁡(3x5)x)y = x \lim_{x \to \infty}\left(\frac{\cot{\left(\frac{3 x}{5} \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⁡(3x5)=−cot⁡(3x5)\cot{\left(\frac{3 x}{5} \right)} = - \cot{\left(\frac{3 x}{5} \right)}
- No
cot⁡(3x5)=cot⁡(3x5)\cot{\left(\frac{3 x}{5} \right)} = \cot{\left(\frac{3 x}{5} \right)}
- No
so, the function
not is
neither even, nor odd
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