Mister Exam
Graphing y = 1/(sh(x)*(ch(x)-1))
The solution
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1
f(x) = ---------------------
sinh(x)*(cosh(x) - 1)
$$f{\left(x \right)} = \frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}}$$
f = 1/((cosh(x) - 1)*sinh(x))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$\frac{\frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} \left(- \left(\cosh{\left(x \right)} - 1\right) \cosh{\left(x \right)} - \sinh^{2}{\left(x \right)}\right)}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} = 0$$
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
$$\frac{\frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} \left(- \left(\cosh{\left(x \right)} - 1\right) \cosh{\left(x \right)} - \sinh^{2}{\left(x \right)}\right)}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} = 0$$
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
$$\frac{\left(\left(\cosh{\left(x \right)} - 1\right) \cosh{\left(x \right)} + \sinh^{2}{\left(x \right)}\right) \left(\frac{\cosh{\left(x \right)}}{\sinh^{2}{\left(x \right)}} + \frac{1}{\cosh{\left(x \right)} - 1}\right) + \frac{\left(\left(\cosh{\left(x \right)} - 1\right) \cosh{\left(x \right)} + \sinh^{2}{\left(x \right)}\right) \cosh{\left(x \right)}}{\sinh^{2}{\left(x \right)}} + \frac{\left(\cosh{\left(x \right)} - 1\right) \cosh{\left(x \right)} + \sinh^{2}{\left(x \right)}}{\cosh{\left(x \right)} - 1} - 4 \cosh{\left(x \right)} + 1}{\left(\cosh{\left(x \right)} - 1\right)^{2} \sinh{\left(x \right)}} = 0$$
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
$$\frac{\left(\left(\cosh{\left(x \right)} - 1\right) \cosh{\left(x \right)} + \sinh^{2}{\left(x \right)}\right) \left(\frac{\cosh{\left(x \right)}}{\sinh^{2}{\left(x \right)}} + \frac{1}{\cosh{\left(x \right)} - 1}\right) + \frac{\left(\left(\cosh{\left(x \right)} - 1\right) \cosh{\left(x \right)} + \sinh^{2}{\left(x \right)}\right) \cosh{\left(x \right)}}{\sinh^{2}{\left(x \right)}} + \frac{\left(\cosh{\left(x \right)} - 1\right) \cosh{\left(x \right)} + \sinh^{2}{\left(x \right)}}{\cosh{\left(x \right)} - 1} - 4 \cosh{\left(x \right)} + 1}{\left(\cosh{\left(x \right)} - 1\right)^{2} \sinh{\left(x \right)}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty} \frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty} \frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty} \frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 1/(sinh(x)*(cosh(x) - 1)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{1}{\cosh{\left(x \right)} - 1} \frac{1}{\sinh{\left(x \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{1}{\cosh{\left(x \right)} - 1} \frac{1}{\sinh{\left(x \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\frac{1}{\cosh{\left(x \right)} - 1} \frac{1}{\sinh{\left(x \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{1}{\cosh{\left(x \right)} - 1} \frac{1}{\sinh{\left(x \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
$$\frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} = - \frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}}$$
- No
$$\frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} = \frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} = - \frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}}$$
- No
$$\frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}} = \frac{1}{\left(\cosh{\left(x \right)} - 1\right) \sinh{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd