3.2 · Critical points, extrema, and the Mean Value Theorem

Find candidates, then compare values

A critical number lies in the domain of $\displaystyle f$ and has $\displaystyle f'(c)=0$ or an undefined derivative. On a closed interval where $\displaystyle f$ is continuous, absolute extrema occur among critical numbers and endpoints. Evaluate $\displaystyle f$, not $\displaystyle f'$, at every candidate.

The Extreme Value Theorem guarantees that a continuous function on a closed bounded interval attains absolute extrema. A missing endpoint or a discontinuity can remove that guarantee.

Use sign changes for local extrema

If $\displaystyle f'$ changes from positive to negative at a critical input, $\displaystyle f$ has a local maximum there. A change from negative to positive gives a local minimum. If no sign change occurs, there need not be a local extremum.

For $\displaystyle f(x)=x^3$, the derivative vanishes at $\displaystyle 0$ but stays nonnegative on both sides. The point is a stationary point without a local maximum or minimum.

Match instantaneous and average rates

If $\displaystyle f$ is continuous on $\displaystyle [a,b]$ and differentiable on $\displaystyle (a,b)$, the Mean Value Theorem guarantees some $\displaystyle c\in(a,b)$ where $\displaystyle f'(c)=(f(b)-f(a))/(b-a)$. State both hypotheses before solving the slope equation.

Rolle’s Theorem is the equal-endpoint case: if also $\displaystyle f(a)=f(b)$, there is a horizontal tangent in the interior. The theorem gives existence, and the equation may have more than one solution.

$$f'(c)=\frac{f(b)-f(a)}{b-a}$$