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Plotting a function graph/ ctg(2*x)-x

Graphing y = ctg(2*x)-x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = cot(2*x) - x
f(x)=−x+cot⁡(2x)f{\left(x \right)} = - x + \cot{\left(2 x \right)} 
f = -x + cot(2*x)
The graph of the function
02468-8-6-4-2-1010-500500
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:
−x+cot⁡(2x)=0- x + \cot{\left(2 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=−9.47734088326452x_{1} = -9.47734088326452
x2=1.8217985837127x_{2} = 1.8217985837127
x3=−1.8217985837127x_{3} = -1.8217985837127
x4=−0.538436993155902x_{4} = -0.538436993155902
x5=0.538436993155902x_{5} = 0.538436993155902
x6=−3.28916686636117x_{6} = -3.28916686636117
x7=6.36114938588332x_{7} = 6.36114938588332
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)−3=0- 2 \cot^{2}{\left(2 x \right)} - 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
8(cot⁡2(2x)+1)cot⁡(2x)=08 \left(\cot^{2}{\left(2 x \right)} + 1\right) \cot{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=π4x_{1} = \frac{\pi}{4}

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
(−∞,π4]\left(-\infty, \frac{\pi}{4}\right]
Convex at the intervals
[π4,∞)\left[\frac{\pi}{4}, \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→−∞(−x+cot⁡(2x))y = \lim_{x \to -\infty}\left(- x + \cot{\left(2 x \right)}\right)
True

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

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