If $f(x)$ is a function such that $f^{\prime}(x)=(x-1)^{2}(4-x),$ then

$A$ cylindrical tank without a top lid is being manufactured to hold a fixed volume of $125\pi \text{ cm}^3$. The minimum surface area required to construct this tank is ............ $\text{cm}^2$. (in $\pi$)

If $S_1$ and $S_2$ are respectively the sets of local minimum and local maximum points of the function $f(x) = 9x^4 + 12x^3 - 36x^2 + 25, x \in R$,then

Point $A$ lies on the curve $y = e^{-x^2}$ and has the coordinates $(x, e^{-x^2})$ where $x > 0$. Point $B$ has the coordinates $(x, 0)$. If $O$ is the origin,then the maximum area of the triangle $AOB$ is

The set of all values of $a$ for which the function $f(x) = (a^2 - 3a + 2) \left( \cos^2 \frac{x}{4} - \sin^2 \frac{x}{4} \right) + (a - 1)x + \sin 1$ does not possess critical points is

$A$ window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is $10 \, m$. Find the dimensions of the window to admit maximum light through the whole opening.