If $f(x) = (\sin^4 x + \cos^4 x)$,$0 < x < \frac{\pi}{2}$,then the function has a minimum value of $ . . . . . . $ at $x = . . . . . . $.

The point $(0, 5)$ is closest to the curve ${x^2} = 2y$ at

Show that the right circular cone of least curved surface area and given volume has an altitude equal to $\sqrt{2}$ times the radius of the base.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x)=|x|+|x^2-1|$. The total number of points at which $f$ attains either a local maximum or a local minimum is

If the function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$,where $a > 0$,attains its maximum and minimum at $p$ and $q$ respectively such that $p^2 = q$,then $a$ equals

Let $S=(-1, \infty)$ and $f: S \rightarrow R$ be defined as $f(x)=\int_{-1}^x (e^t-1)^{11}(2t-1)^5(t-2)^7(t-3)^{12}(2t-10)^{61} dt$. Let $p$ be the sum of the squares of the values of $x$ where $f(x)$ attains local maxima on $S$,and $q$ be the sum of the values of $x$ where $f(x)$ attains local minima on $S$. Then,the value of $p^2+2q$ is