$A$ square piece of tin of side $30\,cm$ is to be made into a box without a top by cutting a square of side $x$ from each corner and folding up the flaps to form a box. If the volume of the box is maximum,then its surface area (in $cm^2$) is equal to $............$.

If a right circular cone having maximum volume is inscribed in a sphere of radius $3 \, cm$,then the curved surface area (in $cm^2$) of this cone is

If $y = a \log |x| + b x^2 + x$ has its extremum values at $x = -1$ and $x = 2$,then

The maximum value of $f(x) = \sin (x)$ in the interval $[-\pi / 2, \pi / 2]$ 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 local maximum value of the function $f(x)=-(x-2)^3(x+2)^2$ is