The Mean Value Theorem and Applications

The Mean Value Theorem (MVT) is the central pillar of differential calculus. While the derivative provides local information about a function’s slope at a single point, the MVT provides the mechanism to translate that local information into global conclusions about the function’s behavior over an interval.

Rolle’s Theorem

The MVT is built upon Rolle’s Theorem. Suppose $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$. If $f(a)=f(b)$, then there exists at least one $c\in (a,b)$ such that $f^{\prime }(c)=0$.

Proof Strategy:

1. By the Extreme Value Theorem, $f$ must attain a maximum $M$ and a minimum $m$ on $[a,b]$.
2. If $M=m$, $f$ is constant, so $f^{\prime }(x)=0$ everywhere.
3. If $M\ne m$, at least one of them must be attained at an interior point $c\in (a,b)$ because $f(a)=f(b)$.
4. At a local extremum $c$ where $f$ is differentiable, Fermat’s Theorem guarantees $f^{\prime }(c)=0$.

The Mean Value Theorem (Lagrange Form)

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists at least one $c\in (a,b)$ such that: $f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$

Geometrically, this means there is a point $c$ where the tangent line is parallel to the secant line passing through $(a,f(a))$ and $(b,f(b))$.

Consequences of the MVT

The MVT allows us to prove several “obvious” properties that are actually quite deep:

1. Constant Function Theorem: If $f^{\prime }(x)=0$ for all $x\in (a,b)$, then $f(x)$ is constant on $(a,b)$.
   • Proof: For any $x_1,x_2$, there exists $c$ such that $f(x_2)-f(x_1)=f^{\prime }(c)(x_2-x_1)=0$.
2. Monotonicity: If $f^{\prime }(x)>0$, then $f$ is strictly increasing. If $f^{\prime }(x)<0$, it is strictly decreasing.
3. Identity Theorem: If $f^{\prime }(x)=g^{\prime }(x)$ for all $x$, then $f(x)=g(x)+C$. This is the foundation of integral calculus.

Cauchy Mean Value Theorem

A generalization involving two functions $f$ and $g$. If $f,g$ are continuous on $[a,b]$ and differentiable on $(a,b)$, there exists $c\in (a,b)$ such that: $(f(b)-f(a))g^{\prime }(c)=(g(b)-g(a))f^{\prime }(c)$ If $g^{\prime }(x)\ne 0$, this can be written as: $\frac{f^{\prime }(c)}{g^{\prime }(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}$ This theorem is the hidden machinery behind the proof of L’Hôpital’s Rule.

Taylor’s Theorem (Mean Value Form)

Taylor’s theorem generalizes the MVT to higher-order derivatives. If $f$ is $n+1$ times differentiable, then: $f(x)=f(a)+f^{\prime }(a)(x-a)+\cdots +\frac{f^{(n)}(a)}{n!}(x-a{)}^n+R_n(x)$ where the remainder $R_n(x)$ can be written in the Lagrange form: $R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a{)}^{n+1}$ for some $c$ between $a$ and $x$. This allows us to put a rigorous bound on the error of a Taylor approximation.

Python: Numerical Verification of MVT

We can use Python and scipy to find the point $c$ predicted by the MVT for a given function.

import numpy as np
from scipy.optimize import brentq
from scipy.misc import derivative

def f(x):
    return x**3 - 3*x + 2

a, b = -1, 2
avg_slope = (f(b) - f(a)) / (b - a)

# Define function for finding f'(c) - avg_slope = 0
def g(c):
    return derivative(f, c, dx=1e-6) - avg_slope

# Use a root finder to find the point c
c = brentq(g, a, b)
print(f"Average Slope: {avg_slope}")
print(f"MVT predicts a point c at: {c}")
print(f"Slope at c: {derivative(f, c, dx=1e-6)}")

Summary

The Mean Value Theorem is the “bridge” of calculus. It allows us to move from the realm of rates (derivatives) to the realm of accumulation and values (functions). Without the MVT, we could not prove that speedometers measure actual distance change, or that a derivative of zero implies a stationary object. It is the logical glue that holds analysis together.

Exercise