If $f(x)$ is a function such that $f^{\prime}(x)=(x-1)^{2}(4-x),$ then

Let $f(x) = x^3 e^{-3x}, x > 0$. Then the maximum value of $f(x)$ is

$A$ manufacturer can sell $x$ items at a price of rupees $\left(5 - \frac{x}{100}\right)$ each. The cost price of $x$ items is $\text{Rs} \left(\frac{x}{5} + 500\right)$. Find the number of items he should sell to earn maximum profit.

Let $f(x)$ be a polynomial of degree $4$ having extreme values at $x = 1$ and $x = 2$. If $\mathop {\lim }\limits_{x \to 0} \left( {\frac{{f(x)}}{{{x^2}}} + 1} \right) = 3$,then $f(-1)$ is equal to

The sum of the absolute minimum and the absolute maximum values of the function $f(x) = |3x - x^2 + 2| - x$ in the interval $[-1, 2]$ is

Show that the semi-vertical angle of a right circular cone of given surface area and maximum volume is $\sin ^{-1}\left(\frac{1}{3}\right)$.