Let $f(x) = \begin{cases} |x| & \text{for } 0 < |x| \leqslant 2 \\ 1 & \text{for } x = 0 \end{cases}$. What is the nature of $f$ at $x = 0$?

The greatest value of the function $f(x) = -1 + \frac{2}{2^{x^2} + 1}$ is:

If $x, y$ are two positive integers such that $x+y=20$ and the maximum value of $x^3 y$ is $k$ at $x=\alpha, y=\beta$,then $\frac{k}{\alpha^2 \beta^2} =$

If $x_0$ is the point of local minima of $f(x) = \bar{a} \cdot (\bar{b} \times \bar{c})$ where $\bar{a} = x \hat{i} - 2 \hat{j} + 3 \hat{k}$,$\bar{b} = -2 \hat{i} + x \hat{j} - \hat{k}$,$\bar{c} = 7 \hat{i} - 2 \hat{j} + x \hat{k}$,then the value of $\bar{a} \cdot \bar{b}$ at $x = x_0$ is

It is given that at $x=1,$ the function $f(x) = x^{4}-62x^{2}+ax+9$ attains its maximum value on the interval $[0,2].$ Find the value of $a.$

The function $f(x)=2 x^3-9 a x^2+12 a^2 x+1$ $(a>0)$ attains its maximum and minimum at $p$ and $q$ respectively and $p^2=q$. Then,$a=$