Mister Exam
Graphing y = arccos(1/lgx)
The solution
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/ 1 \
f(x) = acos|------|
\log(x)/
$$f{\left(x \right)} = \operatorname{acos}{\left(\frac{1}{\log{\left(x \right)}} \right)}$$
f = acos(1/log(x))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 1$$
$$x_{1} = 1$$
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:
$$\operatorname{acos}{\left(\frac{1}{\log{\left(x \right)}} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = e$$
Numerical solution
$$x_{1} = 2.71828182845905$$
so we need to solve the equation:
$$\operatorname{acos}{\left(\frac{1}{\log{\left(x \right)}} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = e$$
Numerical solution
$$x_{1} = 2.71828182845905$$
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
$$\frac{1}{x \sqrt{1 - \frac{1}{\log{\left(x \right)}^{2}}} \log{\left(x \right)}^{2}} = 0$$
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
$$\frac{1}{x \sqrt{1 - \frac{1}{\log{\left(x \right)}^{2}}} \log{\left(x \right)}^{2}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
Vertical asymptotes
Have:
$$x_{1} = 1$$
$$x_{1} = 1$$
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:
$$\operatorname{acos}{\left(\frac{1}{\log{\left(x \right)}} \right)} = \operatorname{acos}{\left(\frac{1}{\log{\left(- x \right)}} \right)}$$
- No
$$\operatorname{acos}{\left(\frac{1}{\log{\left(x \right)}} \right)} = - \operatorname{acos}{\left(\frac{1}{\log{\left(- x \right)}} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\operatorname{acos}{\left(\frac{1}{\log{\left(x \right)}} \right)} = \operatorname{acos}{\left(\frac{1}{\log{\left(- x \right)}} \right)}$$
- No
$$\operatorname{acos}{\left(\frac{1}{\log{\left(x \right)}} \right)} = - \operatorname{acos}{\left(\frac{1}{\log{\left(- x \right)}} \right)}$$
- No
so, the function
not is
neither even, nor odd