The number of distinct real roots of $x^{4}-4x+1=0$ is

Find two numbers whose sum is $24$ and whose product is as large as possible.

Let $f(x)=(x+3)^2(x-2)^3$ for $x \in [-4,4]$. If $M$ and $m$ are the maximum and minimum values of $f$ respectively in $[-4,4]$,then the value of $M-m$ is:

Let $x_0$ be the point of local minima of $f(x) = \overline{a} \cdot (\overline{b} \times \overline{c})$ where $\overline{a} = x \hat{i} - 2 \hat{j} + 3 \hat{k}$,$\overline{b} = -2 \hat{i} + x \hat{j} - \hat{k}$,and $\overline{c} = 7 \hat{i} - 2 \hat{j} + x \hat{k}$. Then the value of $\overline{a} \cdot \overline{b}$ at $x = x_0$ is:

The height of the cone of maximum volume inscribed in a sphere of radius $R$ is

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