Mean Value Theorems: Proofs, Consequences, and Extensions

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State the Mean Value Theorem.

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

How is Rolle’s theorem a special case of the Mean Value Theorem?

When $f(a)=f(b)$, the endpoint secant slope is zero, so the MVT gives an interior point where $f'(c)=0$.

What auxiliary function proves the Mean Value Theorem using Rolle’s theorem?

For $r=\frac{f(b)-f(a)}{b-a}$, use $g(x)=f(x)-rx$; its endpoint values are equal.

What conclusion does the scalar multivariable mean value theorem give along a line segment?

For some point $z$ on the segment from $x$ to $y$, $f(y)-f(x)=\nabla f(z)\cdot(y-x)$.

What does the integral mean value theorem guarantee for continuous $f$ on $[a,b]$?

Some $c\in(a,b)$ satisfies $\int_a^b f(x)\,dx=f(c)(b-a)$.

Quiz questions

1. Which equation is the general cross-multiplied conclusion of Cauchy’s Mean Value Theorem for continuous $f,g$ on $[a,b]$, differentiable on $(a,b)$?Answer: For some $c\in(a,b)$, $(f(b)-f(a))g'(c)=(g(b)-g(a))f'(c)$.
2. For differentiable scalar-valued $f$ on a domain containing the segment from $x$ to $y$, what identity does the multivariable Mean Value Theorem guarantee?Answer: For some point $z$ on the segment, $f(y)-f(x)=\nabla f(z)\cdot(y-x)$.
3. Why does the scalar Mean Value Theorem’s exact equality generally fail for vector-valued functions?Answer: Different components may satisfy their scalar equalities at different points on the segment.
4. If $f$ is continuous on $[a,b]$, what does the integral mean value theorem guarantee?Answer: Some $c\in(a,b)$ satisfies $\int_a^b f(x)\,dx=f(c)(b-a)$.
5. For $f(x)=|x|$ on $[-1,1]$, why can the standard Mean Value Theorem not supply a point with derivative equal to the endpoint secant slope?Answer: The secant slope is zero, but the derivative is never zero where it exists.