Consider the function $f\left(x\right)=-2x^2+4x+1$ on the interval $[-1,2]$. Find the critical points of $f$ and use the First Derivative Test to classify these points. Then, determine the absolute maximum and minimum values of $f$ on the specified interval (if there are any).

Consider the function $f\left(x\right)=-2x\sqrt{x+12}$. Determine intervals where $f$ is increasing and decreasing, and the intervals on which the curve is concave up and concave down.

Find all critical points and domain endpoints for the function $y=x^3-192\sqrt{x}$.

A ball is thrown vertically upward from the ground. The height of the ball, in meters, $t$ seconds after it has been thrown, is given by $y=-10g{t}^2+ut$. Given that $g=10{\text{ m/s}}^2$ and $u=50\text{ m/s}$, calculate the maximum height reached by the ball.

Determine the intervals on which the function $h$ is increasing or decreasing, given its derivative $h^{\prime }\left(x\right)=(x+15)(x+21)(x-13)$.

For the following cubic function, how many local extreme values (local minima or maxima) does $f$ have?

$f\left(x\right)=x^3-6x^2+9x+7$