Unit 05 · Term 1

The Mean Value Theorem & Inverse Functions

The MVT is the workhorse that turns information about $f'$ into information about $f$. Almost every "derivative tells you about the function" fact — monotonicity, constancy, inequalities, and differentiability of inverse functions — comes from it.

By the end you can

State and prove Rolle's theorem and the Mean Value Theorem.
Prove that $f'=0$ on an interval forces $f$ constant.
Relate the sign of $f'$ to increasing/decreasing behaviour.
Use the one-variable inverse function theorem to differentiate inverse functions.
Prove inequalities with the MVT.

Why it matters & where it comes from

Why it exists

The Mean Value Theorem is the bridge from a derivative at one instant to behaviour over a whole stretch. It is why "if your speed never topped 100, your average can't have either" is a theorem, not just a feeling.

In the real world

Average-speed cameras on highways literally issue tickets using the MVT — the time between two points gives an average speed the law treats as proof you sped somewhere. The same theorem underlies error bounds in numerical methods and uniqueness proofs for differential equations.

Story & open questions

It is named partly for Rolle (1691), who was himself an early skeptic of calculus. Cauchy gave it its modern form. Simple as it looks, almost every "the derivative controls the function" fact in this course flows from this one theorem.

What this unit changes

Before the Mean Value Theorem, a derivative is local information: it describes behavior at one point. The MVT turns that local information into a statement about an entire interval. That is why it becomes the proof engine for monotonicity, inequalities, uniqueness, and "same derivative means same function up to a constant."

MVT proof pattern

Choose two points $a<b$ in the interval.

Check $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$.

Use $f(b)-f(a)=f'(c)(b-a)$ for some $c\in(a,b)$.

Insert information about $f'(c)$ to conclude something about $f(b)-f(a)$.

Typical proof goals

Many MVT problems look different but share the same skeleton. Learn to identify the target before doing computations.

Show increasing. Prove $f(x_2)-f(x_1)>0$ for arbitrary $x_1<x_2$.
Show constant. Prove $f(x_2)-f(x_1)=0$ for arbitrary $x_1<x_2$.
Prove an inequality. Apply MVT to the difference between the two sides, or bound $f'(c)$ directly.
Show uniqueness of a root. Combine IVT for existence with monotonicity or Rolle's theorem for at most one root.

Local extrema and Fermat's theorem

Theorem Fermat

If $f$ has a local extremum at an interior point $c$ and $f'(c)$ exists, then $f'(c)=0$.

Such points (where $f'=0$ or $f'$ fails to exist) are the critical points — the only candidates for interior extrema.

Rolle's theorem

Theorem Rolle

If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then there is $c\in(a,b)$ with $f'(c)=0$.

Proof idea

By the Extreme Value Theorem $f$ attains a max and min on $[a,b]$.

If the maximum equals the minimum, $f$ is constant and $f'=0$ everywhere. Otherwise, because $f(a)=f(b)$, at least one extremum is interior, and Fermat gives $f'(c)=0$. ∎

The Mean Value Theorem

Theorem MVT

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there is $c\in(a,b)$ with $$f'(c)=\frac{f(b)-f(a)}{b-a}.$$

Proof idea

Subtract the secant line: let $g(x)=f(x)-\big[f(a)+ \tfrac{f(b)-f(a)}{b-a}(x-a)\big]$.

Then $g(a)=g(b)=0$; Rolle gives $g'(c)=0$, which rearranges to the MVT. ∎

Consequences

Corollaries

On an interval: if $f'=0$ everywhere then $f$ is constant; if $f'>0$ then $f$ is strictly increasing; if $f'=g'$ then $f-g$ is constant.

Example An inequality

Show $e^x\ge1+x$ for all $x$.

Let $h(x)=e^x-1-x$, so $h(0)=0$ and $h'(x)=e^x-1$.

$h'>0$ for $x>0$ and $h'<0$ for $x<0$, so $x=0$ is a global minimum; thus $h(x)\ge0$. ∎

The inverse function theorem in one variable

The inverse function theorem is where the MVT becomes a practical tool for new derivative formulas. Once a differentiable function is known to be one-to-one and its derivative is not zero, the inverse has a derivative whose value is forced by the chain rule.

Theorem inverse derivative rule

Let $f$ be continuous and strictly monotone on an interval $I$, and suppose $f$ is differentiable at $a\in I$ with $f'(a)\ne0$. If $b=f(a)$, then $f^{-1}$ is differentiable at $b$ and $$(f^{-1})'(b)=\frac1{f'(a)}=\frac1{f'(f^{-1}(b))}.$$

Proof idea

For $y=f(x)$ and $b=f(a)$, write the inverse difference quotient as $$\frac{f^{-1}(y)-f^{-1}(b)}{y-b}=\frac{x-a}{f(x)-f(a)}.$$

As $y\to b$, strict monotonicity makes $x\to a$. The right side tends to $1/f'(a)$ because $f'(a)\ne0$.

Course connection. This theorem justifies derivative formulas for $\ln x$, $\arctan x$, and other inverse functions after the original function is known to be one-to-one. The MVT supplies the one-to-one condition through monotonicity.

Common pitfalls

• The MVT needs continuity on the closed interval and differentiability on the open one — check both.
• The MVT guarantees $c$ exists but not its value or uniqueness.
• To use the inverse derivative rule, identify the original input $a=f^{-1}(b)$; do not plug the output $b$ into $f'$ by accident.
• "$f'>0$ at a point" does not make $f$ increasing near that point in general; the clean statement is about $f'>0$ on a whole interval.

Practice

Problems adapted from the MAT137 question bank by Alfonso Gracia-Saz, used under CC BY-SA 4.0. Write full solutions before opening the answer.

Practice 1 · Zero derivative

Use the MVT to prove: if $f'(x)=0$ for all $x\in(a,b)$, then $f$ is constant on $(a,b)$.

Solution

For any $x_1<x_2$ in the interval, MVT gives $f(x_2)-f(x_1)=f'(c)(x_2-x_1)=0$, so $f(x_2)=f(x_1)$.

Practice 2 · Lipschitz bound

Prove $|\sin a-\sin b|\le|a-b|$ for all real $a,b$.

Hint

Apply the MVT to $\sin$ on $[a,b]$: the difference equals $\cos c\,(a-b)$, and $|\cos c|\le1$.

Practice 3 · Inequality

Show $\ln(1+x)<x$ for all $x>0$.

Hint

Let $h(x)=x-\ln(1+x)$; $h(0)=0$ and $h'(x)=1-\tfrac1{1+x}>0$ for $x>0$, so $h$ increases.

Practice 4 · Bounding

If $f'(x)\le M$ on $[a,b]$, bound $f(b)-f(a)$.

Answer

MVT gives $f(b)-f(a)=f'(c)(b-a)\le M(b-a)$.

Practice 5 · Exponential bound

Prove that $e^x\ge1+x$ for all $x\in\mathbb{R}$.

Solution

Let $g(x)=e^x-1-x$. Then $g(0)=0$ and $g'(x)=e^x-1$, which is $<0$ for $x<0$ and $>0$ for $x>0$. So $g$ has a global minimum $0$ at $x=0$; hence $g(x)\ge0$.

Practice 6 · Counting roots

Prove that a polynomial of degree $n$ has at most $n$ real zeros.

Solution

If it had $n+1$ distinct zeros, Rolle's theorem between consecutive zeros gives $n$ zeros of the derivative, then $n-1$ of the second derivative, and so on, forcing the degree-$n$ leading term ($n!\,a_n\ne0$) to have a zero — impossible.

Practice 7 · One-to-one

Prove: if $f$ is differentiable on an open interval $I$ and $f'(x)\ne0$ for all $x\in I$, then $f$ is one-to-one on $I$.

Solution

If $f(x_1)=f(x_2)$ with $x_1<x_2$, Rolle's theorem gives $c\in(x_1,x_2)$ with $f'(c)=0$, contradicting the hypothesis. So $f$ is one-to-one.

Practice 8 · A derivative at 0

Suppose $f$ is differentiable at $0$ with $f(0)=0$ and $f(x)\ge x$ for all $x$. Find $f'(0)$.

Solution

For $x>0$, $\tfrac{f(x)}{x}\ge1$; for $x<0$, dividing by a negative flips it to $\tfrac{f(x)}{x}\le1$. The two one-sided limits are both $f'(0)$, so $f'(0)\ge1$ and $f'(0)\le1$, giving $f'(0)=1$.

Practice 9 · Inverse derivative

Let $f(x)=2x+x^3$. Prove $f$ has an inverse on $\mathbb{R}$ and compute $(f^{-1})'(0)$.

Solution

$f'(x)=2+3x^2>0$ for all $x$, so MVT gives strict increase on every interval and hence injectivity. Also $f(0)=0$ and $f'(0)=2$, so $(f^{-1})'(0)=1/f'(0)=1/2$.

Practice 10 · Inverse trig

Use the inverse function theorem to prove $\dfrac{d}{dx}\arctan x=\dfrac1{1+x^2}$.

Solution

Let $y=\arctan x$, so $x=\tan y$ with $y\in(-\pi/2,\pi/2)$. Since $(\tan)'y=\sec^2y\ne0$, the inverse derivative is $1/\sec^2y=\cos^2y=1/(1+\tan^2y)=1/(1+x^2)$.

Expanded topic checklist

Course coverage. This unit is part of the complete MAT137 topic map. It should cover the following items before you move on:

Rolle and MVT. Use compactness through EVT, then Rolle, then MVT as the main proof sequence.
Derivative consequences. Prove constant, increasing, decreasing, and inequality results from derivative sign information.
Inverse functions. Use monotonicity and the inverse derivative rule to justify derivatives of inverse functions.
Root counting. Use Rolle repeatedly to control how many roots a differentiable function or polynomial can have.

Textbook/source alignment

Use the Alfonso Gracia-Saz MAT137 materials as the primary local spine, with the current UofT single-variable calculus topic split as the completeness check. The added items are original study prompts, not copied textbook exercises.

Core theorem map

The spine is: extrema give stationary points, stationary points give Rolle, Rolle gives MVT, and MVT converts derivative hypotheses into global conclusions.

Study routine. For each problem, decide first whether the goal is existence of a point, monotonicity, an inequality, or an inverse derivative; that choice selects the theorem.

Deep practice extensions

Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.

Extension 1 · Inequality

Use the MVT to prove $|\cos a-\cos b|\le |a-b|$ for all real $a,b$.

Guide

Apply MVT to $\cos x$ on the interval between $a$ and $b$; the difference is $-\sin(c)(a-b)$ and $|\sin c|\le1$.

Extension 2 · Inverse rule

If $f(2)=5$ and $f\prime(2)=7$, what is $(f^{-1})\prime(5)$ assuming the inverse theorem applies?

Guide

The inverse derivative at output $5$ is $1/f\prime(2)=1/7$.

Extension 3 · Monotonicity

Prove that if $f\prime(x)>0$ on an interval, then $f$ is strictly increasing there.

Guide

For $x_1<x_2$, MVT gives $f(x_2)-f(x_1)=f\prime(c)(x_2-x_1)>0$.

Extension 4 · Rolle chain

Explain why a differentiable function with three distinct zeros must have at least two distinct zeros of its derivative.

Guide

Apply Rolle on the two intervals between consecutive zeros; the points are in disjoint open intervals, so they are distinct.

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