Differentiation - Main Theorem

The Mean Value Theorem is one of the most important theoretical results in differential calculus. It bridges local information (the derivative at a point) with global information (the average rate of change over an interval).

TheoremRolle's Theorem

Let $f$ be a function that satisfies:

$f$ is continuous on $[a, b]$
$f$ is differentiable on $(a, b)$
$f(a) = f(b)$

Then there exists at least one point $c \in (a, b)$ such that $f'(c) = 0$ .

Geometrically, Rolle's Theorem says that if a continuous, differentiable function returns to its starting value, then somewhere in between, the tangent line must be horizontal.

ExampleApplying Rolle's Theorem

Show that $f(x) = x^3 - 3x + 2$ has at least one critical point in $(-2, 2)$ .

Note that $f(-2) = -8 + 6 + 2 = 0$ and $f(2) = 8 - 6 + 2 = 4$ . Since $f(-2) \neq f(2)$ , we cannot directly apply Rolle's Theorem.

However, we can verify that $f(-1) = -1 + 3 + 2 = 4 = f(2)$ . Since $f$ is a polynomial (hence continuous and differentiable everywhere), Rolle's Theorem guarantees there exists $c \in (-1, 2)$ with $f'(c) = 0$ .

Indeed, $f'(x) = 3x^2 - 3 = 0$ when $x = \pm 1$ , so $c = 1 \in (-1, 2)$ is such a point.

TheoremMean Value Theorem (MVT)

Let $f$ be a function that satisfies:

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

Then there exists at least one point $c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$

The MVT states that at some point, the instantaneous rate of change equals the average rate of change. Geometrically, there exists a point where the tangent line is parallel to the secant line connecting $(a, f(a))$ and $(b, f(b))$ .

Remark

Rolle's Theorem is a special case of the MVT where $f(a) = f(b)$ , so the average rate of change is zero. The MVT can be proven by applying Rolle's Theorem to an auxiliary function.

ExampleUsing the Mean Value Theorem

Suppose a car travels 100 miles in 2 hours. By the MVT, at some moment during the trip, the car's instantaneous velocity must have been exactly $\frac{100}{2} = 50$ mph.

More formally, if $s(t)$ represents position at time $t$ , with $s(0) = 0$ and $s(2) = 100$ , then there exists $c \in (0, 2)$ such that: $s'(c) = \frac{s(2) - s(0)}{2 - 0} = \frac{100}{2} = 50$

TheoremConsequences of the MVT

The Mean Value Theorem has several important consequences:

If $f'(x) = 0$ for all $x \in (a, b)$ , then $f$ is constant on $(a, b)$
If $f'(x) = g'(x)$ for all $x \in (a, b)$ , then $f$ and $g$ differ by a constant
If $f'(x) > 0$ for all $x \in (a, b)$ , then $f$ is strictly increasing on $(a, b)$
If $f'(x) < 0$ for all $x \in (a, b)$ , then $f$ is strictly decreasing on $(a, b)$

These consequences form the foundation for curve sketching and optimization. The MVT connects the sign of the derivative to monotonicity, allowing us to determine where functions increase or decrease.