Find the intervals where the function $g(x) = (2x - 3)^2$ is increasing and decreasing.

Determine the $x$-coordinate of the critical point of the function $f\left(x\right)=2x\sqrt{x-2b}$, where $b$ is a constant and $b>0$.

Find the critical points of the function $f\left(x\right)=\cos \left(3x\right)-2$ on the interval $[-\frac{\pi }{2},\frac{\pi }{2}]$. Identify the absolute maximum and minimum values.

On the interval $[-2,3]$, find the absolute maximum and minimum of the function $f\left(x\right)=2x^5-20x^3+50x$.

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

Determine the intervals of concavity and the inflection points of the function using the graph of $f^{\prime \prime }\left(x\right)$.

Given a function $f\left(x\right)$ defined on the interval $[-3, 5.5],$ write the intervals where the function $f\left(x\right)$ is concave up.

Locate the critical points of $f\left(x\right)=3x^{-4}-x^{-3}$, and use the Second Derivative Test to identify whether these points are local maxima or minima.

A function $f\left(x\right)$ has the following properties:

$f^{\prime}\left(x\right)>0$ and $f^{\prime}$$^{\prime}\left(x\right)>0$, for

Which of the following is a possible graph of $f\left(x\right)$ for ?

Graph the function $f(x)=3e^{-\frac{x^2}{3}}$.

$f'(x) = -2x e^{-x^2/3}$

$f''(x) = e^{-x^2/3} \left(\frac{4x^2}{3} - 2 \right)$

A rectangle has its base on the $x$-axis and two vertices on the curve $y=100-4x^2$. Determine the length $l$ and width $w$ that yield the maximum area. Then, find the maximum area.

A company plans to manufacture lampshades in the shape of a cone from circular pieces of fabric with a radius of $15\text{ in}$. What are the dimensions of the cone with the maximum volume that can be produced?

A rectangular swimming pool is being designed with an area of $150{\text{ m}}^2$. Surrounding the pool on all sides is a uniform $2\text{ m}$ wide deck. What are the dimensions of the pool that minimize the total area (including the deck)?