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Plotting a function graph/ log(cot(2*x))

Graphing y = log(cot(2*x))

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = log(cot(2*x))
f(x)=log⁡(cot⁡(2x))f{\left(x \right)} = \log{\left(\cot{\left(2 x \right)} \right)} 
f = log(cot(2*x))
The graph of the function
02468-8-6-4-2-1010-1010
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(cot(2*x)).
log⁡(cot⁡(0⋅2))\log{\left(\cot{\left(0 \cdot 2 \right)} \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
−2cot⁡2(2x)−2cot⁡(2x)=0\frac{- 2 \cot^{2}{\left(2 x \right)} - 2}{\cot{\left(2 x \right)}} = 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
4(−(cot⁡2(2x)+1)2cot⁡2(2x)+2cot⁡2(2x)+2)=04 \left(- \frac{\left(\cot^{2}{\left(2 x \right)} + 1\right)^{2}}{\cot^{2}{\left(2 x \right)}} + 2 \cot^{2}{\left(2 x \right)} + 2\right) = 0
Solve this equation
The roots of this equation
x1=−π8x_{1} = - \frac{\pi}{8}
x2=π8x_{2} = \frac{\pi}{8}

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
[−π8,π8]\left[- \frac{\pi}{8}, \frac{\pi}{8}\right]
Convex at the intervals
(−∞,−π8]∪[π8,∞)\left(-\infty, - \frac{\pi}{8}\right] \cup \left[\frac{\pi}{8}, \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→−∞log⁡(cot⁡(2x))y = \lim_{x \to -\infty} \log{\left(\cot{\left(2 x \right)} \right)}
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞log⁡(cot⁡(2x))y = \lim_{x \to \infty} \log{\left(\cot{\left(2 x \right)} \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(cot(2*x)), 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→−∞(log⁡(cot⁡(2x))x)y = x \lim_{x \to -\infty}\left(\frac{\log{\left(\cot{\left(2 x \right)} \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(log⁡(cot⁡(2x))x)y = x \lim_{x \to \infty}\left(\frac{\log{\left(\cot{\left(2 x \right)} \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:
log⁡(cot⁡(2x))=log⁡(−cot⁡(2x))\log{\left(\cot{\left(2 x \right)} \right)} = \log{\left(- \cot{\left(2 x \right)} \right)}
- No
log⁡(cot⁡(2x))=−log⁡(−cot⁡(2x))\log{\left(\cot{\left(2 x \right)} \right)} = - \log{\left(- \cot{\left(2 x \right)} \right)}
- No
so, the function
not is
neither even, nor odd
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