The maximum volume of the right circular cone with slant height $6$ units is

Let $f : (-\infty, \infty) \to (-\infty, \infty)$ be defined by $f(x) = x^3 + 1$.
Statement-$1$: The function has a local extremum at $x = 0$.
Statement-$2$: The function $f(x)$ is continuous and differentiable on $(-\infty, \infty)$ and $f'(0) = 0$.

Prove that the function $h(x) = x^{3} + x^{2} + x + 1$ does not have any local maxima or local minima.

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

Local maximum and local minimum values of the function $f(x) = (x - 1)(x + 2)^2$ are

If $m$ and $M$ respectively denote the minimum and maximum of $f(x)=(x-1)^2+3$ for $x \in [-3, 1]$,then the ordered pair $(m, M)$ is equal to