When designing a cylindrical water storage tank, an engineer is tasked with maximizing the volume for a fixed amount of surface area (representing material cost). The volume function $V(r)$ is defined for the radius $r$ on a closed interval $[r_{min}, r_{max}]$, dictated by physical site constraints. Because $V(r)$ is a continuous polynomial function on a closed interval, the Extreme Value Theorem (EVT) guarantees that an absolute maximum volume must exist. The engineer knows they only need to check the critical points where $V'(r)=0$ and the boundary endpoints $r_{min}$ and $r_{max}$ to find the optimal design.

Automated toll systems on tollways record the exact time a vehicle enters and exits the highway. If the distance between two toll booths is 120 kilometers, and a driver completes the journey in exactly 1 hour, their average speed is 120 km/h. If the posted speed limit is 100 km/h, the Mean Value Theorem (MVT) provides mathematical proof that at least one specific instant during that hour, the driver's exact speedometer reading (instantaneous velocity) was exactly 120 km/h. Law enforcement agencies use this principle for point-to-point average speed cameras.

Find the absolute maximum and minimum values of the function $f(x) = x^3 - 3x^2 + 1$ on the closed interval $[-1/2, 4]$.

Find the absolute extrema of $f(x) = x - 2\sin x$ on the interval $[0, 2\pi]$.

Why does the Extreme Value Theorem fail for $f(x) = 1/x$ on the interval $[-1, 1]$?

Verify Rolle's Theorem for the function $f(x) = x^2 - 4x + 3$ on the interval $[1, 3]$.

Verify Rolle's Theorem for $f(x) = \sin x$ on the interval $[0, \pi]$.

Why does Rolle's Theorem fail for $f(x) = |x|$ on the interval $[-1, 1]$?

Apply the Mean Value Theorem to the function $f(x) = x^3 - x$ on the interval $[0, 2]$.

Apply the MVT to the function $f(x) = \ln(x)$ on the interval $[1, e]$.

Show that the equation $x^5 + 10x + 3 = 0$ has exactly one real root using Rolle's Theorem and IVT.

Verify Cauchy's Mean Value Theorem for the functions $f(x) = x^2$ and $g(x) = x^3$ on the interval $[1, 2]$.

Use Cauchy's Mean Value Theorem to prove the first step of L'Hopital's Rule for limits resulting in $0/0$.