Applications of Derivatives - Main Theorem

The relationship between monotonicity and the first derivative is fundamental to understanding function behavior. These theorems formalize how the sign of the derivative determines whether a function is increasing or decreasing.

TheoremMonotonicity and the First Derivative

Let $f$ be continuous on $[a, b]$ and differentiable on $(a, b)$ .

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 $[a, b]$
If $f'(x) = 0$ for all $x \in (a, b)$ , then $f$ is constant on $[a, b]$

The proof relies on the Mean Value Theorem: for any $x_1 < x_2$ in $[a, b]$ , there exists $c \in (x_1, x_2)$ with $f(x_2) - f(x_1) = f'(c)(x_2 - x_1)$

If $f'(c) > 0$ , then $f(x_2) > f(x_1)$ , establishing that $f$ is increasing.

ExampleProving a Function is Increasing

Show that $f(x) = x + \sin x$ is strictly increasing on $\mathbb{R}$ .

$f'(x) = 1 + \cos x$

Since $-1 \leq \cos x \leq 1$ , we have $0 \leq 1 + \cos x \leq 2$ for all $x$ .

Actually, $f'(x) = 0$ only when $\cos x = -1$ , which occurs at isolated points $x = \pi + 2\pi k$ . Between these points, $f'(x) > 0$ , and by continuity, $f$ is (non-strictly) increasing everywhere and strictly increasing on any interval not containing these isolated points.

TheoremFirst Derivative Test (Precise Statement)

Let $f$ be continuous at a critical point $c$ .

If $f'(x) > 0$ for $x$ in some interval $(a, c)$ and $f'(x) < 0$ for $x$ in some interval $(c, b)$ , then $f$ has a local maximum at $c$
If $f'(x) < 0$ for $x$ in some interval $(a, c)$ and $f'(x) > 0$ for $x$ in some interval $(c, b)$ , then $f$ has a local minimum at $c$
If $f'$ has the same sign on both sides of $c$ , then $f$ has no local extremum at $c$

ExampleApplying Monotonicity

Find all intervals where $g(x) = x^3 - 3x^2 - 9x + 5$ is increasing or decreasing.

$g'(x) = 3x^2 - 6x - 9 = 3(x^2 - 2x - 3) = 3(x - 3)(x + 1)$

Critical points: $x = -1, 3$

Sign analysis:

$x < -1$ : $g'(x) = 3(−)(−) = + > 0$ (increasing)
$-1 < x < 3$ : $g'(x) = 3(−)(+) = − < 0$ (decreasing)
$x > 3$ : $g'(x) = 3(+)(+) = + > 0$ (increasing)

Therefore: increasing on $(-\infty, -1] \cup [3, \infty)$ , decreasing on $[-1, 3]$ .

TheoremRolle's Theorem Between Consecutive Roots

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

Corollary: Between any two consecutive roots of a differentiable function, there must be at least one critical point.

ExampleApplication to Polynomials

The polynomial $p(x) = x^3 - 3x + 1$ has three real roots. How many critical points must it have?

$p'(x) = 3x^2 - 3 = 3(x^2 - 1) = 3(x-1)(x+1)$

Critical points: $x = -1, 1$ (exactly two).

By Rolle's Theorem, between each pair of consecutive roots, there must be at least one critical point. With three roots, there are two gaps between them, requiring at least two critical points—which matches our calculation.

These theorems transform qualitative geometric intuition about increasing and decreasing functions into precise mathematical statements with rigorous proofs.