Mister Exam
Graphing y = x*sh((10-10log(x))/10)
The solution
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/10 - 10*log(x)\
f(x) = x*sinh|--------------|
\ 10 /
$$f{\left(x \right)} = x \sinh{\left(\frac{10 - 10 \log{\left(x \right)}}{10} \right)}$$
f = x*sinh((10 - 10*log(x))/10)
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:
$$x \sinh{\left(\frac{10 - 10 \log{\left(x \right)}}{10} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - e$$
$$x_{2} = e$$
Numerical solution
$$x_{1} = -2.71828182845905$$
$$x_{2} = 2.71828182845905$$
so we need to solve the equation:
$$x \sinh{\left(\frac{10 - 10 \log{\left(x \right)}}{10} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - e$$
$$x_{2} = e$$
Numerical solution
$$x_{1} = -2.71828182845905$$
$$x_{2} = 2.71828182845905$$
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
$$- \frac{\sinh{\left(\log{\left(x \right)} - 1 \right)} + \cosh{\left(\log{\left(x \right)} - 1 \right)}}{x} = 0$$
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
$$- \frac{\sinh{\left(\log{\left(x \right)} - 1 \right)} + \cosh{\left(\log{\left(x \right)} - 1 \right)}}{x} = 0$$
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
$$\lim_{x \to -\infty}\left(x \sinh{\left(\frac{10 - 10 \log{\left(x \right)}}{10} \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x \sinh{\left(\frac{10 - 10 \log{\left(x \right)}}{10} \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(x \sinh{\left(\frac{10 - 10 \log{\left(x \right)}}{10} \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x \sinh{\left(\frac{10 - 10 \log{\left(x \right)}}{10} \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x*sinh((10 - 10*log(x))/10), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty} \sinh{\left(\frac{10 - 10 \log{\left(x \right)}}{10} \right)} = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \sinh{\left(\frac{10 - 10 \log{\left(x \right)}}{10} \right)} = -\infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \sinh{\left(\frac{10 - 10 \log{\left(x \right)}}{10} \right)} = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \sinh{\left(\frac{10 - 10 \log{\left(x \right)}}{10} \right)} = -\infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
$$x \sinh{\left(\frac{10 - 10 \log{\left(x \right)}}{10} \right)} = - x \sinh{\left(1 - \log{\left(- x \right)} \right)}$$
- No
$$x \sinh{\left(\frac{10 - 10 \log{\left(x \right)}}{10} \right)} = x \sinh{\left(1 - \log{\left(- x \right)} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$x \sinh{\left(\frac{10 - 10 \log{\left(x \right)}}{10} \right)} = - x \sinh{\left(1 - \log{\left(- x \right)} \right)}$$
- No
$$x \sinh{\left(\frac{10 - 10 \log{\left(x \right)}}{10} \right)} = x \sinh{\left(1 - \log{\left(- x \right)} \right)}$$
- No
so, the function
not is
neither even, nor odd