Graph the function $g(x)=x\sqrt[\phantom{2}]{9-x^2}$ by finding the following:

$($a$)$ Domain

$($b$)$ Critical numbers

$($c$)$ Regions where $g$ is increasing$/$decreasing

$($d$)$ Max's$/$min$\text{'}$s

$($e$)$ Concavity

$($f$)$ Inflection points

A piece of wire $10$ m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is as large as possible?

Use the mean value theorem to show that $f(x)=2x+cos(x)$ has only one $x-$intercept.

A plane flying with a constant speed of $300k\frac{m}{h}r$ passes over a ground radar station at an altitude of $1$ km and climbs at an angle of $30.$ At what rate is the distance from the plane to the radar station increasing a minute later?

Let $f(x)$ be a continuous function on $a,b$ and differentiable on $(a,b).$ Suppose $m{f}^{\text{'}}(x)M$ for all xin$(a,b)($in this case, we say $f^{\text{'}}(x)$ is bounded$),$ where $m$ and $M$ are constants.

$($a$)$ Prove that $m(b-a)f(b)-f(a)M(b-a).$

$($b$)$ Using this result, explain how $f^{\text{'}}(x)$ being bounded affects the "steepness" of the function $f(x)$ on $a,b.$