Mister Exam
Graphing y = log(cot(2*x))
The solution
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f(x) = log(cot(2*x))
$$f{\left(x \right)} = \log{\left(\cot{\left(2 x \right)} \right)}$$
f = log(cot(2*x))
The graph of the function
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{- 2 \cot^{2}{\left(2 x \right)} - 2}{\cot{\left(2 x \right)}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{- 2 \cot^{2}{\left(2 x \right)} - 2}{\cot{\left(2 x \right)}} = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$4 \left(- \frac{\left(\cot^{2}{\left(2 x \right)} + 1\right)^{2}}{\cot^{2}{\left(2 x \right)}} + 2 \cot^{2}{\left(2 x \right)} + 2\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{8}$$
$$x_{2} = \frac{\pi}{8}$$
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
$$\left[- \frac{\pi}{8}, \frac{\pi}{8}\right]$$
Convex at the intervals
$$\left(-\infty, - \frac{\pi}{8}\right] \cup \left[\frac{\pi}{8}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$4 \left(- \frac{\left(\cot^{2}{\left(2 x \right)} + 1\right)^{2}}{\cot^{2}{\left(2 x \right)}} + 2 \cot^{2}{\left(2 x \right)} + 2\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\pi}{8}$$
$$x_{2} = \frac{\pi}{8}$$
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
$$\left[- \frac{\pi}{8}, \frac{\pi}{8}\right]$$
Convex at the intervals
$$\left(-\infty, - \frac{\pi}{8}\right] \cup \left[\frac{\pi}{8}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \log{\left(\cot{\left(2 x \right)} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \log{\left(\cot{\left(2 x \right)} \right)}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \log{\left(\cot{\left(2 x \right)} \right)}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \log{\left(\cot{\left(2 x \right)} \right)}$$
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
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\log{\left(\cot{\left(2 x \right)} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\log{\left(\cot{\left(2 x \right)} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\log{\left(\cot{\left(2 x \right)} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\log{\left(\cot{\left(2 x \right)} \right)}}{x}\right)$$
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:
$$\log{\left(\cot{\left(2 x \right)} \right)} = \log{\left(- \cot{\left(2 x \right)} \right)}$$
- No
$$\log{\left(\cot{\left(2 x \right)} \right)} = - \log{\left(- \cot{\left(2 x \right)} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\log{\left(\cot{\left(2 x \right)} \right)} = \log{\left(- \cot{\left(2 x \right)} \right)}$$
- No
$$\log{\left(\cot{\left(2 x \right)} \right)} = - \log{\left(- \cot{\left(2 x \right)} \right)}$$
- No
so, the function
not is
neither even, nor odd