M171 Calculus I · Unit 4: Optimization & the integral · on Exam 4
§3.3 Global optimization
Study notes for Global optimization in M171 Calculus I at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- Extreme Value Theorem: a continuous function on a closed interval $[a,b]$ attains an absolute maximum and minimum.
- Closed interval method: find critical numbers in $(a,b)$, evaluate $f$ at those and at the endpoints $a$, $b$; the largest value is the absolute max, the smallest the absolute min.
- On an open or infinite interval, use the shape of $f$ (sign of $f'$, limits at the ends) instead: a single critical number where $f'$ changes from $-$ to $+$ is a global min.
Key formulas
Candidates
$$\{a,\ b\}\ \cup\ \{c\in(a,b): f'(c)=0 \text{ or undefined}\}$$
Worked example
Find the absolute extremes of $f(x) = x^3 - 3x$ on $[0,3]$.
Show the solution
$f'(x) = 3x^2-3 = 0 \Rightarrow x = \pm1$; only $x=1$ is inside. $f(0)=0$, $f(1)=-2$, $f(3)=18$. Absolute max $18$ at $x=3$, absolute min $-2$ at $x=1$.
Common mistakes
- Forgetting the endpoints.
- Including critical numbers that lie outside the interval.
Exam tipMake a three-column table: candidate x, f(x), and why it is a candidate.
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