The maximum value of the function $f(x) = 3x^3 - 18x^2 + 27x - 40$ on the set $S = \{x \in R : x^2 + 30 \leq 11x\}$ is

Find the local maximum and local minimum values for the function $f(x) = x\sqrt{1 - x}$ where $0 < x < 1$.

The local maximum value of the function $f(x)=-(x-2)^3(x+2)^2$ is

$A$ stone thrown upwards has its equation of motion $s = 490t - 4.9t^2$. Then the maximum height reached by it is:

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,$a=$

$A$ 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:
$(A)$ $24$
$(B)$ $32$
$(C)$ $45$
$(D)$ $60$