Mister Exam
Graphing y = x^((tgx)(ctgx))
The solution
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tan(x)*cot(x) f(x) = x
$$f{\left(x \right)} = x^{\tan{\left(x \right)} \cot{\left(x \right)}}$$
f = x^(tan(x)*cot(x))
The graph of the function
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:
$$x^{\tan{\left(x \right)} \cot{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0$$
so we need to solve the equation:
$$x^{\tan{\left(x \right)} \cot{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0$$
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
$$x^{\tan{\left(x \right)} \cot{\left(x \right)}} \left(\left(\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \left(- \cot^{2}{\left(x \right)} - 1\right) \tan{\left(x \right)}\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right) = 0$$
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
$$x^{\tan{\left(x \right)} \cot{\left(x \right)}} \left(\left(\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} + \left(- \cot^{2}{\left(x \right)} - 1\right) \tan{\left(x \right)}\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right) = 0$$
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
$$x^{\tan{\left(x \right)} \cot{\left(x \right)}} \left(\left(\left(\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right)^{2} + 2 \left(- \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) + \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \cot{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \cot{\left(x \right)}\right) \log{\left(x \right)} + \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} + \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{x} - \frac{\left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} - \frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x^{2}}\right) = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$x^{\tan{\left(x \right)} \cot{\left(x \right)}} \left(\left(\left(\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x}\right)^{2} + 2 \left(- \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) + \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \cot{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} \cot{\left(x \right)}\right) \log{\left(x \right)} + \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} + \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{x} - \frac{\left(\cot^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} - \frac{\tan{\left(x \right)} \cot{\left(x \right)}}{x^{2}}\right) = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
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:
$$y = \lim_{x \to -\infty} x^{\tan{\left(x \right)} \cot{\left(x \right)}}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} x^{\tan{\left(x \right)} \cot{\left(x \right)}}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} x^{\tan{\left(x \right)} \cot{\left(x \right)}}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} x^{\tan{\left(x \right)} \cot{\left(x \right)}}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x^(tan(x)*cot(x)), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{x^{\tan{\left(x \right)} \cot{\left(x \right)}}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{x^{\tan{\left(x \right)} \cot{\left(x \right)}}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{x^{\tan{\left(x \right)} \cot{\left(x \right)}}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{x^{\tan{\left(x \right)} \cot{\left(x \right)}}}{x}\right)$$
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:
$$x^{\tan{\left(x \right)} \cot{\left(x \right)}} = \left(- x\right)^{\tan{\left(x \right)} \cot{\left(x \right)}}$$
- No
$$x^{\tan{\left(x \right)} \cot{\left(x \right)}} = - \left(- x\right)^{\tan{\left(x \right)} \cot{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$x^{\tan{\left(x \right)} \cot{\left(x \right)}} = \left(- x\right)^{\tan{\left(x \right)} \cot{\left(x \right)}}$$
- No
$$x^{\tan{\left(x \right)} \cot{\left(x \right)}} = - \left(- x\right)^{\tan{\left(x \right)} \cot{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd