The maximum area of a rectangle that can be inscribed in a circle of radius $2 \text{ unit}$ is (in square unit)

The number of critical points of the function $f(x)=(x-2)^{2/3}(2x+1)$ 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:
$(A)$ $24$
$(B)$ $32$
$(C)$ $45$
$(D)$ $60$

Let $f: D \rightarrow R$, $D \subseteq R$, $c \in D$ and $r$ be a non-zero real number. Consider the following statements:
$Y$. $c$ is an extreme point of $f \Rightarrow c$ is an extreme point of $rf$
$M$. $c$ is an extreme point of $f \Rightarrow c$ is an extreme point of $r+f$
Which of the following is correct?

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

Twenty metres of wire is available to fence off a flower bed in the form of a circular sector. What must the radius of the circle be, if the area of the flower bed is to be the greatest (in $m$)?