The function $S(x) = \int\limits_0^x {\sin \left( {\frac{{\pi {t^2}}}{2}} \right)\,dt} $ has two critical points in the interval $[1, 2.4]$. One of the critical points is a local minimum and the other is a local maximum. The local minimum occurs at $x =$

The function $f(x) = \int_1^x \{ 2(t - 1)(t - 2)^3 + 3(t - 1)^2(t - 2)^2 \} dt$ is maximum when $x$ is equal to

Show that the function given by $f(x) = \frac{\log x}{x}$ has a maximum at $x = e$.

The shortest distance between the point $\left( \frac{3}{2}, 0 \right)$ and the curve $y = \sqrt{x}, (x > 0)$ is:

$A$ box is to be made with a square base and an open top. If the area of the material used is $48 \, m^2$,then the maximum volume of the box is ........... $m^3$.

$A$ 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