Application of Derivatives Questions

A spherical iron ball $10\text{ cm}$ in radius is coated with a layer of ice of uniform thickness that melts at a rate of $50{\text{ cm}}^3\text{/min}$. When the thickness of ice is $5\text{ cm}$, then the rate at which the thickness of ice decreases, is:

Given $P(x)=x^4+a{x}^3+b{x}^2+cx+d$ such that $x=0$ is the only real root of $P^{\prime }(x)=0$. If $P(-1)<P(1)$, then in the interval $[-1,1]$:

A spherical balloon is filled with $4500\pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72\pi$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases $49$ minutes after the leakage began is:

Let $a,b\in \mathbb{R}$ be such that the function $f$ given by $f(x)=\ln |x|+b{x}^2+ax,x\ne 0$ has extreme values at $x=-1$ and $x=2$.Statement 1: $f$ has local maximum at $x=-1$ and at $x=2$.Statement 2: $a=\frac{1}{2}$ and $b=-\frac{1}{4}$.