13. Law of the Mean · Increasing and Decreasing Functions

Adapted from: Classical single-variable calculus (relative extrema, Rolle, Mean Value Theorem / Law of the Mean, monotone from $f'$). Rewritten for teaching with interactive explorers — not a verbatim textbook dump.

This lesson ties geometry (horizontal tangents, parallel chords) to tests you use constantly: where can local max/min occur, and when is a function increasing or decreasing?

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Relative maximum and minimum

A function $f$ has a relative (local) maximum at $x_0$ if

$$f(x_0)\ge f(x)$$

for every $x$ in some open interval around $x_0$ (where $f$ is defined). A relative minimum flips the inequality. Either is a relative extremum.

Local, not global

Relative extrema only compare nearby points. A local max need not be the tallest point on the whole graph.

Theorem — critical point

If $f$ has a relative extremum at $x_0$ and $f'(x_0)$ exists, then $f'(x_0)=0$.

So a differentiable local max/min has a horizontal tangent. (Nondifferentiable corners like $|x|$ at $0$ can still be extrema without a derivative.)

In Fig. 13-1, horizontal tangents appear at $A$ (relative maximum) and $B$ (relative minimum).

Fig. 13-1

Check: relative extrema

Quiz

Relative extrema

1 / 2

If f is differentiable at a relative maximum x₀, then…

Pick an answer to see if you are right.

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Rolle’s theorem

Rolle. Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$, with $f(a)=f(b)=0$. Then there is at least one $x_0\in(a,b)$ with $f'(x_0)=0$.

Geometrically: if the graph hits the $x$-axis at both ends and is smooth in between, some interior tangent is horizontal (Fig. 13-2).

Fig. 13-2

Generalized Rolle. Same continuity/differentiability, but only $g(a)=g(b)$. Then $g'(x_0)=0$ for some interior point (Fig. 13-3). Proof idea: apply Rolle to $f(x)=g(x)-g(a)$.

Fig. 13-3

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Mean Value Theorem (Law of the Mean)

MVT. If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then some $x_0\in(a,b)$ satisfies

$$\frac{f(b)-f(a)}{b-a}=f'(x_0).$$

The left side is the secant (chord) slope between $P_1=(a,f(a))$ and $P_2=(b,f(b))$. The right side is a tangent slope. So somewhere the tangent is parallel to $P_1P_2$ (Fig. 13-4).

Fig. 13-4

This is also called the Law of the Mean.

Extended / higher-order forms (preview)

If $f$ and $g$ are continuous on $[a,b]$, differentiable on $(a,b)$, and $g'\ne 0$ on $(a,b)$, then for some $x_0$

$$\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f'(x_0)}{g'(x_0)}.$$

(The ordinary MVT is $g(x)=x$.)

A higher-order law of the mean expands $f(b)$ like a Taylor polynomial of degree $n-1$ about $a$, with remainder $\dfrac{f^{(n)}(x_0)}{n!}(b-a)^n$ for some $x_0$ between $a$ and $b$. Specializing $b\mapsto x$ or $a=0$ recovers the usual Taylor remainder forms.

Check: MVT

Quiz

Mean Value Theorem

1 / 2

Geometrically, the conclusion of the MVT is that…

Pick an answer to see if you are right.

Hands On: Find an MVT point

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Increasing and decreasing functions

On an interval, $f$ is increasing if $u<v$ implies $f(u)<f(v)$, and decreasing if $u<v$ implies $f(u)>f(v)$.

Theorem — sign of $f'$

• If $f'>0$ on an interval, then $f$ is increasing there.
• If $f'<0$ on an interval, then $f$ is decreasing there.

Proof sketch: for $u<v$, MVT gives $f(v)-f(u)=f'(c)(v-u)$ with the same sign as $f'(c)$.

Check: monotone

Quiz

Monotone from f′

1 / 1

If f′(x) = 2 + cos x, then on ℝ the function f is…

Pick an answer to see if you are right.

Hands On: Classify from f′

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Key takeaways

• Differentiable relative extrema occur where $f'=0$ (horizontal tangent).
• Rolle / generalized Rolle: equal endpoint values ⇒ an interior horizontal tangent.
• MVT (Law of the Mean): average rate equals some instantaneous rate — tangent parallel to chord.
• Sign of $f'$ decides increasing vs decreasing on an interval.

Further Reading

• 14. Maximum and Minimum Values — next: critical numbers and derivative tests
• 13b. Law of the Mean — Practice — next: worksheets + guided proofs
• Any standard calculus text — Rolle’s theorem and the Mean Value Theorem chapter
• 5.1 Differentiation algorithms — numerical cousins of $f'$
• Taylor remainder forms — higher-order law of the mean in disguise