Consider a cuboidal 8 m³ container with a square base and no top: each side of the base measures a, and the height of the container is b. Sketch the graph of the function that depicts the surface area of the container S(a) for a > 0.

Using the graph, estimate the value of a that minimizes the surface area, and round your answer to 2 decimal places.

Determine the critical points of the function $f\left(x\right)=\frac{x}{x^2+4}$.

A projectile is launched vertically upwards and its height above the ground after $t$ seconds is given by the function $h(t) = 50t - 5t^2$, where $0\(\le\) t<10$. At what time is the projectile at its maximum height?

Calculate the critical points for the function $p\left(x\right)=2x^4-16x^2+32$ on the interval $[-2,3]$. Identify the absolute maximum and minimum values.

Check if the function $f\left(x\right)=2x\sqrt{6-x}$ satisfies the conditions of the following theorem on its domain. If it does, identify the location and the value of the absolute extremum guaranteed by the theorem.

Theorem: Suppose $f$ is continuous on an interval $I$ that contains exactly one local extremum at $c$. If a local maximum occurs at $c$, then $f\left(c\right)$ is the absolute maximum of $f$ on $I$. If a local minimum occurs at $c$, then $f\left(c\right)$ is the absolute minimum of $f$ on $I$.

Determine the absolute extreme values of the function.

Given the function $f\left(x\right)=\mid x^4-8x\mid$, does $f^{\prime }\left(2\right)$ exist?

Determine the intervals on which the function $h$ is increasing or decreasing given its derivative $h^{\prime }\left(x\right)=1-\frac{25}{x^2}$, $x\ne 0$.

Which of the following graphs has $f^{\prime }\left(x\right)>0$ at $(-3,0)$ and $(3,\infty )$, and $f^{\prime }\left(x\right)<0$ at $(-\infty ,-3)$ and $(0,3)$?

Determine the intervals on which the following function is concave up or concave down. Identify any inflection points.

$k\left(x\right)=\sqrt[3]{x+5}$

Analyze the concavity of the quadratic function $h\left(x\right)=m{x}^2+nx+p$. Determine the conditions on $m$, $n$, and $p$ for $h\left(x\right)$ to be concave up and concave down.