Extrema and optimization

Finding Local Extrema

A critical point occurs where $f'(c) = 0$ or $f'(c)$ is undefined. This is our primary tool for locating peaks and valleys.

The first derivative test tracks the sign of $f'$ around the critical point to classify its nature:

The second derivative test evaluates $f''(c)$ when $f'(c) = 0$:

$$f''(c) > 0 \;\Rightarrow\; \text{local minimum} \qquad f''(c) < 0 \;\Rightarrow\; \text{local maximum}$$

Common pitfall: $f'(c) = 0$ makes $c$ a candidate, not a guarantee. For example, $f(x) = x^3$ has a derivative of zero at $0$, but pauses flat mid-climb and is neither a maximum nor a minimum.

Global Extrema & MVT

To find global extrema on $[a,b]$, evaluate $f$ at every critical point and at both endpoints. The largest resulting value is the absolute maximum, and the smallest is the absolute minimum. True extremes often hide at the endpoints where the derivative never vanishes.

The Mean Value Theorem (MVT) states that if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, there exists at least one $c \in (a,b)$ such that:

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

This guarantees a point where the instantaneous rate of change equals the average rate of change over the interval.

Physics link: Minimising potential energy $U(x)$ to find the equilibrium position is a direct application where $U'(x) = 0$.