AP Calculus BC / Unit 5: Analytical Applications of Differentiation
5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
Notes
- Extreme Value Theorem: a continuous function on a closed interval $[a,b]$ has both an absolute max and an absolute min.
- Critical point: $x=c$ in the domain where $f'(c)=0$ or $f'(c)$ doesn't exist.
- Local extrema in the interior happen only at critical points.
Practice
Critical points of $f(x)=x^3-3x$ are at:
$3x^2-3=0$.
The EVT guarantees a max and min when $f$ is:
Both conditions, continuous and closed, are needed.
$f(x)=x^{2/3}$ has a critical point at $x=0$ because:
$\displaystyle f'=\frac{2}{3}x^{-1/3}$ is undefined at 0, and 0 is in the domain.
Ready for more? Take the Unit 5 test.