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

Graphing y = ctg(x/3)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          /x\
f(x) = cot|-|
          \3/
f(x)=cot⁡(x3)f{\left(x \right)} = \cot{\left(\frac{x}{3} \right)} 
f = cot(x/3)
The graph of the function
05-20-15-10-5101520-250250
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⁡(x3)=0\cot{\left(\frac{x}{3} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=3π2x_{1} = \frac{3 \pi}{2}
Numerical solution
x1=23.5619449019235x_{1} = 23.5619449019235
x2=−4.71238898038469x_{2} = -4.71238898038469
x3=−23.5619449019235x_{3} = -23.5619449019235
x4=98.9601685880785x_{4} = 98.9601685880785
x5=61.261056745001x_{5} = 61.261056745001
x6=4.71238898038469x_{6} = 4.71238898038469
x7=−51.8362787842316x_{7} = -51.8362787842316
x8=−32.9867228626928x_{8} = -32.9867228626928
x9=80.1106126665397x_{9} = 80.1106126665397
x10=51.8362787842316x_{10} = 51.8362787842316
x11=−89.5353906273091x_{11} = -89.5353906273091
x12=42.4115008234622x_{12} = 42.4115008234622
x13=89.5353906273091x_{13} = 89.5353906273091
x14=−80.1106126665397x_{14} = -80.1106126665397
x15=−42.4115008234622x_{15} = -42.4115008234622
x16=−98.9601685880785x_{16} = -98.9601685880785
x17=−61.261056745001x_{17} = -61.261056745001
x18=−70.6858347057703x_{18} = -70.6858347057703
x19=70.6858347057703x_{19} = 70.6858347057703
x20=−14.1371669411541x_{20} = -14.1371669411541
x21=32.9867228626928x_{21} = 32.9867228626928
x22=14.1371669411541x_{22} = 14.1371669411541
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cot(x/3).
∞~\tilde{\infty}
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
−cot⁡2(x3)3−13=0- \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
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
2(cot⁡2(x3)+1)cot⁡(x3)9=0\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
x1=3π2x_{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
(−∞,3π2]\left(-\infty, \frac{3 \pi}{2}\right]
Convex at the intervals
[3π2,∞)\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→−∞cot⁡(x3)=⟨−∞,∞⟩\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=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞cot⁡(x3)=⟨−∞,∞⟩\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=⟨−∞,∞⟩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→−∞(cot⁡(x3)x)=lim⁡x→−∞(cot⁡(x3)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)
Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(cot⁡(x3)x)y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{x}{3} \right)}}{x}\right)
lim⁡x→∞(cot⁡(x3)x)=lim⁡x→∞(cot⁡(x3)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)
Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(cot⁡(x3)x)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⁡(x3)=−cot⁡(x3)\cot{\left(\frac{x}{3} \right)} = - \cot{\left(\frac{x}{3} \right)}
- No
cot⁡(x3)=cot⁡(x3)\cot{\left(\frac{x}{3} \right)} = \cot{\left(\frac{x}{3} \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = ctg(x/3)
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