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

The maximum value of $xy$ subject to $x+y=7$ is

$A$ rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8:15$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is $100$,the resulting box has maximum volume. The lengths of the sides of the rectangular sheet are:
$(A)$ $24$
$(B)$ $32$
$(C)$ $45$
$(D)$ $60$

Local maximum value of the function $\frac{\log x}{x}$ is

The function $f(x) = \sin^p x \cos^q x$ has a maximum at:

Find the total number of local maxima and local minima of the function $f(x) = \begin{cases} (2+x)^3, & -3 < x \leq -1 \\ x^{2/3}, & -1 < x < 2 \end{cases}$