Where is a quantity largest or smallest? The graph of f' answers this: at a smooth peak or valley the tangent line is flat. Combined with the The Chain Rule, we can handle almost any function.
Key idea
A critical number of $f$ is an $x = c$ in the domain where $f'(c) = 0$ or $f'(c)$ does not exist. Closed interval method: a continuous $f$ on $[a, b]$ has an absolute maximum and minimum (Extreme Value Theorem); they occur at critical numbers or at the endpoints. Evaluate $f$ at all of them and compare.
Why it works
Fermat's theorem: if $f$ has a local max at an interior point $c$ and $f'(c)$ exists, then $f'(c) = 0$. Reason: near a max, $f(c + h) - f(c) \le 0$. For $h > 0$ the difference quotient is $\le 0$; for $h < 0$ it is $\ge 0$. Both one-sided limits equal $f'(c)$, so $f'(c) \le 0$ and $f'(c) \ge 0$: it must be 0. So interior extremes can only happen at critical numbers, and the only other candidates on $[a, b]$ are the endpoints.
Worked example: Closed interval method
Find the absolute max and min of $f(x) = x^3 - 3x$ on $[-2, 3]$.
- $f'(x) = 3x^2 - 3 = 3(x - 1)(x + 1)$: critical numbers $\pm 1$, both inside.
- Evaluate: $f(-2) = -2$, $f(-1) = 2$, $f(1) = -2$, $f(3) = 18$.
- Largest: 18. Smallest: $-2$ (at two places).
Answer: Max 18 at $x = 3$; min $-2$ at $x = -2$ and $x = 1$.
Worked example: Where the derivative fails
Find the critical numbers of $f(x) = x^{2/3}(x - 5)$.
- $f(x) = x^{5/3} - 5x^{2/3}$, so $$f'(x) = \frac{5}{3}x^{2/3} - \frac{10}{3}x^{-1/3} = \frac{5x - 10}{3x^{1/3}}$$.
- $f'(x) = 0$ when $x = 2$.
- $f'(0)$ does not exist (division by 0), but $f(0) = 0$ is defined.
Answer: Critical numbers: $0$ (a cusp) and $2$.
Common mistake
Two errors. First, assuming every critical number is a max or min: $f(x) = x^3$ has $f'(0) = 0$, yet the graph keeps rising through 0. Second, forgetting the endpoints, which is exactly where the max of the first example sits. Check: on a closed interval, list critical numbers and endpoints before comparing.
A local (relative) max is higher than nearby points; an absolute max is the highest point on the whole interval. An endpoint can be an absolute extreme without the tangent being flat.