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 $f\left(x\right)=\frac{3x^2}{x^2-9}$ on the interval $[-5,5]$. 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.