M171 Calculus I · Unit 3: Implicit, limits & extrema · on Exam 3
§3.1 Using derivatives to identify extreme values
Study notes for Using derivatives to identify extreme values 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 StudyCat.
Big ideas
- A critical number $c$ is a point in the domain where $f'(c)=0$ or $f'(c)$ is undefined. Local extrema of a continuous function can only occur at critical numbers (but not every critical number is an extremum).
- First derivative test: if $f'$ changes from $+$ to $-$ at $c$, local max; from $-$ to $+$, local min; no sign change, neither.
- Second derivative test: if $f'(c)=0$ and $f''(c)<0$, local max; $f''(c)>0$, local min; $f''(c)=0$, inconclusive (use the first derivative test).
- Inflection points occur where $f''$ changes sign (candidates: $f''=0$ or undefined).
Key formulas
Critical numbers
$$f'(c) = 0 \;\text{ or }\; f'(c)\ \text{undefined}$$
Second derivative test
$$f'(c)=0,\ f''(c)<0 \Rightarrow \text{local max};\qquad f''(c)>0 \Rightarrow \text{local min}$$
Worked example
Classify the critical numbers of $f(x) = 2x^3 - 9x^2 + 12x$.
Show the solution
$f'(x) = 6x^2 - 18x + 12 = 6(x-1)(x-2)$, critical at $1$ and $2$. Sign chart: $f'>0$ on $(-\infty,1)$, $<0$ on $(1,2)$, $>0$ on $(2,\infty)$. Local max at $x=1$ ($f(1)=5$), local min at $x=2$ ($f(2)=4$). Check: $f''(x) = 12x-18$, $f''(1)=-6<0$, $f''(2)=6>0$.
Common mistakes
- Reporting the critical number when the question wants the extreme value $f(c)$, or vice versa.
- Assuming $f''(c)=0$ means an inflection point without checking for a sign change.
Exam tipA sign chart with test points is the fastest complete answer; it also earns explanation credit.
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