The maximum value of $f(x) = \frac{x}{4 + x + x^2}$ on $[-1, 1]$ is

The function $f(x) = x\sqrt{1 - x^2}$ for $x > 0$ has:

Statement-$I$: $e^{\pi} > \pi^e$.
Statement-$II$: The function $f(x) = x^{1/x}$ attains its global maximum at $x = e$.

$A$ cylindrical container is to be made from a certain solid material with the following constraints: It has a fixed inner volume of $V \ mm^3$,has a $2 \ mm$ thick solid wall,and is open at the top. The bottom of the container is a solid circular disc of thickness $2 \ mm$ and has a radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is $10 \ mm$,then the value of $\frac{V}{250 \pi}$ is

Find the absolute maximum and minimum values of the function $f$ given by $f(x) = \cos^{2} x + \sin x$,where $x \in [0, \pi]$.

Prove that the function $g(x) = \log x$ does not have any maxima or minima.