14. Maximum and Minimum Values

Adapted from: Classical single-variable calculus (critical numbers, first/second derivative tests, absolute extrema on closed intervals). Rewritten for teaching — not a verbatim textbook dump.

After Law of the Mean, we hunt where extrema occur and how to classify them.

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Critical numbers

A number $x_0$ in the domain of $f$ is a critical number if either

$$f'(x_0)=0\qquad\text{or}\qquad f'(x_0)\text{ is undefined}.$$

From Theorem 13.1: a differentiable relative extremum must occur at a critical number ($f'=0$). The converse fails — e.g. $f(x)=x^{3}$ has $f'(0)=0$ but neither a relative max nor min at $0$.

Examples

$f$	$f'$	Critical numbers
$7x^{2}-3x+5$	$14x-3$	$\dfrac{3}{14}$
$x^{3}-2x^{2}+x+1$	$3x^{2}-4x+1$	$1$ and $\dfrac{1}{3}$
$x^{2/3}$	$\dfrac{2}{3}x^{-1/3}$	$0$ ($f'$ undefined)

Check: critical numbers

Quiz

Critical numbers

1 / 1

Which statement is true?

Pick an answer to see if you are right.

Hands On: Find critical numbers

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Second derivative test

Assume $f'(x_0)=0$ and $f''(x_0)$ exists.

$f''(x_0)$	Conclusion
$<0$	relative maximum
$>0$	relative minimum
$=0$	inconclusive

When $f''(x_0)=0$, anything can happen: $x^{4}$ (min), $-x^{4}$ (max), $x^{3}$ (neither) — all have $f''(0)=0$.

Examples

• $f(x)=7x^{2}-3x+5$: critical $\dfrac{3}{14}$, $f''=14>0$ → relative min.
• $f(x)=x^{3}-2x^{2}+x+1$: $f''=6x-4$; at $1$ get $+2$ (min), at $\dfrac{1}{3}$ get $-2$ (max).
• $f(x)=x^{2/3}$: critical $0$ with $f'$ undefined → second-derivative test N/A.

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First derivative test

Useful when $f''(x_0)=0$, $f''$ is hard, or you prefer sign charts. Assume $f'(x_0)=0$ (or a critical number where a sign chart still makes sense).

Signs of $f'$ left → right of $x_0$	Conclusion
$\{+,-\}$	relative maximum
$\{-,+\}$	relative minimum
$\{+,+\}$ or $\{-,-\}$	neither

Fig. 14-1(a) {+, −} max

Fig. 14-1(b) {−, +} min

Fig. 14-1(c) {+, +} neither

Fig. 14-1(d) {−, −} neither

Example — when $f''=0$

At $0$ for $x^{4}$, $-x^{4}$, $x^{3}$:

• $4x^{3}$: $\{-,+\}$ → relative min
• $-4x^{3}$: $\{+,-\}$ → relative max
• $3x^{2}$: $\{+,+\}$ → neither (inflection)

Check: first vs second test

Quiz

Derivative tests

1 / 2

f′(c)=0 and f″(c)=0. What next?

Pick an answer to see if you are right.

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Absolute maximum and minimum

On a set $S$, an absolute maximum at $x_0\in S$ means $f(x)\le f(x_0)$ for all $x\in S$; absolute minimum flips the inequality.

Tabular method on $[a,b]$

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, the Extreme Value Theorem guarantees abs max/min. Find them by:

1. Critical numbers $c_i$ in $(a,b)$
2. Table: $a$, each $c_i$, and $b$
3. Evaluate $f$ — largest value = abs max; smallest = abs min

Fig. 14-2 — absolute extrema on a closed interval

Example

$f(x)=x^{3}-x^{2}-x+2$ on $[0,2]$.

$f'(x)=(3x+1)(x-1)$ → critical numbers $-\dfrac{1}{3}$ (outside) and $1$ (inside).

$x$	$f(x)$
$0$	$2$
$1$	$1$
$2$	$4$

Absolute maximum $4$ at $x=2$; absolute minimum $1$ at $x=1$.

Hands On: Absolute extrema table

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Unique relative extremum ⇒ absolute

Theorem. If continuous $f$ on an interval $J$ (finite or infinite) has a unique relative extremum in $J$, that extremum is also absolute on $J$.

Fig. 14-4 — a lone relative max is absolute on the interval

Idea: if some $d$ beat the unique relative max at $c$, EVT on the segment between $c$ and $d$ would force another interior extremum — contradiction. For a unique relative min, apply the result to $-f$.

Check: absolute extrema

Quiz

Absolute extrema

1 / 1

On [a,b], why include endpoints in the candidate table?

Pick an answer to see if you are right.

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Key takeaways

• Critical numbers: $f'=0$ or $f'$ undefined — candidates for relative extrema.
• Second-derivative test: sign of $f''(c)$ when $f'(c)=0$; inconclusive if $f''(c)=0$.
• First-derivative test: sign patterns $\{+,-\}$, $\{-,+\}$, same-sign.
• Absolute on $[a,b]$: compare $f$ at endpoints and interior critical numbers.
• A unique relative extremum on an interval is absolute there.

Further Reading

• 15. Curve Sketching · Concavity · Symmetry — next: assemble the full graph
• 14b. Max/Min — Practice — solved-problem drills (extrema + optimization)
• 13. Law of the Mean — relative extrema and $f'=0$
• 13b. Law of the Mean — Practice — Rolle / MVT drills
• Any standard calculus text — related rates / further applied optimization