If $x - 2y = 4,$ the minimum value of $xy$ is

If $f(x)=\sin x+2 \cos ^{2} x$ for $\frac{\pi}{4} \leq x \leq \frac{3 \pi}{4}$, then $f$ attains its

Find the maximum and the minimum values,if any,of the function $f$ given by $f(x) = x^{2}, x \in R$.

Let $V_1$ be the volume of a given right circular cone with $O$ as the centre of the base and $A$ as its apex. Let $V_2$ be the maximum volume of the right circular cone inscribed in the given cone whose apex is $O$ and whose base is parallel to the base of the given cone. Then,the ratio $V_2 / V_1$ is

The smaller side of the rectangle with the largest area, that can be inscribed inside a semi-circle of radius $2 \ units$ is of length

If $x = 1$ is a critical point of the function $f(x) = (3x^{2} + ax - 2 - a)e^{x}$,then