Consider the quartic function $h\left(x\right)$ with its second derivative given by $h^{\prime \prime }\left(x\right)=(x+6)(x-5)$. Find the $x$-values where the graph of $h\left(x\right)$ has an inflection point.

Locate the critical points of $f\left(x\right)=e^x{(2x-3)}^2$, and use the Second Derivative Test to identify whether these points are local maxima, minima, or neither.

Consider the function $g\left(x\right)$ whose first derivative is given by $g^{\prime }\left(x\right)=x^2-2x-15$. Determine the $x$-coordinate at which $g\left(x\right)$, if any, has a local maximum, local minimum, or inflection point.

Determine the points where the function $h$ has local maxima or minima given $h^{\prime }\left(x\right)=(x-15)(x+21)(x-13)$.

Mark owns an electric vehicle with a dashboard monitor that displays the driving range based on battery charge. The number of miles you can drive with $c$ kWh of battery charge remaining is given as $r\left(c\right)=60c-30.5c^2+14.2c^3-2.1c^4$ for $0\le c\le 3.5$. Graph the function using a calculator and interpret the driving range function $r\left(c\right)$.

A container is leaking oil, and the volume of oil left in the container after $t$ days can be modeled by the function $V=200(90-t{)}^2$, where $V$ is measured in liters. Graph the volume function using a graphing utility and calculate the initial volume of oil in the container before it started to leak.

A damped harmonic oscillator's displacement from equilibrium is described by the equation $x\left(t\right)=4e^{-0.5t}\cos \left(4t\right)$, where $t$ is time in seconds. Find the first time at which the oscillator's displacement is at a maximum.

A thermal storage chest with a square base and a box-like shape must hold $20{\text{ ft}}^3$ of material. The bottom panel is made of reinforced insulation, which is three times more expensive per square foot than the side panels, while the top lid is constructed from a lightweight material that costs the same as the sides. Determine the side length $s$ of the square base and the height $h$ of the storage chest that will minimize the total material cost.

An engineer is designing a cylindrical water tank to hold $5000{\text{ cm}}^3$ of water. Determine the radius of the tank that will minimize the surface area.