On the interval $[0, 1],$ the function $f(x) = {x^{25}}{(1 - x)^{75}}$ takes its maximum value at the point

$A$ 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 $x=-1$ and $x=2$ be the critical points of the function $f(x)=x^3+ax^2+b \ln|x|+1, x \neq 0$. Let $m$ and $M$ respectively be the absolute minimum and the absolute maximum values of $f$ in the interval $\left[-2, -\frac{1}{2}\right]$. Then $|M+m|$ is equal to (Take $\ln 2 \approx 0.7$):

The perimeter of a sector is $p$. The area of the sector is maximum when its radius is

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

The maximum value $M$ of $f(x) = 3^x + 5^x - 9^x + 15^x - 25^x$,as $x$ varies over all real numbers,satisfies: