Extrema and optimization

Extrema and Optimisation

A critical point occurs where $f'(c) = 0$ or $f'(c)$ is undefined.

First derivative test — Track the sign of $f'$:

Second derivative test — When $f'(c)=0$:

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

Global extrema on $[a,b]$ — Evaluate $f$ at every critical point and at both endpoints; the largest value is the absolute maximum, the smallest the absolute minimum.

Mean Value Theorem

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, there exists $c\in(a,b)$ with

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

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

Common pitfall: $f'(c) = 0$ makes $c$ a candidate, not a winner: $x^{3}$ pauses flat at 0 mid-climb. And on closed intervals, the true extremes may hide at the endpoints, where the derivative never vanishes.