Increasing and Decreasing Functions: Critical Numbers and Intervals

Section 5.1: Increasing and Decreasing Functions

Introduction

This section explores how to determine where a function is increasing or decreasing, and introduces the concept of critical numbers. These ideas are foundational for analyzing the behavior of functions, especially in business calculus where optimization and trend analysis are essential.

Definitions

• Increasing Function: A function f is increasing over an interval I if, for every a and b in I, if a < b, then f(a) < f(b).

• Decreasing Function: A function f is decreasing over an interval I if, for every a and b in I, if a < b, then f(a) > f(b).

Examples:

• Increasing on $(-\infty, \infty)$: The function is always rising as x increases.

• Decreasing on $(0,2)$: The function falls as x increases from 0 to 2.

• Increasing on $(-\infty, 1)$, Decreasing on $(1, \infty)$: The function rises until x = 1, then falls.

• Increasing on $(-\infty, -3) \cup (-1, \infty)$, Decreasing on $(-3, -1)$: The function rises, falls, then rises again.

Critical Numbers and Critical Points

Changes from increasing to decreasing (or vice versa) often occur at special points:

• Where the tangent line is horizontal (derivative is zero)

• Where there is a sharp point (derivative does not exist)

Definition: A critical number (or critical value) of a function f is any number c in the domain of f for which $f'(c) = 0$ or $f'(c)$ does not exist. The point $(c, f(c))$ is called a critical point.

• Example: Find all critical numbers for $f(x) = 2x^2 - 8x + 12$.

  • Compute the derivative: $f'(x) = 4x - 8$

  • Set $f'(x) = 0$: $4x - 8 = 0 \implies x = 2$

  • There are no points where $f'(x)$ does not exist (since it is a polynomial).

  • Critical number: $x = 2$

Test for Intervals of Increase and Decrease

To determine where a function is increasing or decreasing, use the sign of the derivative:

• If $f'(x) > 0$ for each $x$ in an interval, $f$ is increasing on that interval.

• If $f'(x) < 0$ for each $x$ in an interval, $f$ is decreasing on that interval.

• If $f'(x) = 0$ for each $x$ in an interval, $f$ is constant on that interval.

Procedure for Finding Intervals of Increase and Decrease

1. Determine the domain of $f$. (For polynomials, the domain is $(-\infty, \infty)$.)

2. Find $f'(x)$.

3. Find all critical numbers by solving $f'(x) = 0$ and where $f'(x)$ does not exist.

4. Make a sign diagram for the first derivative:

   1. Draw a number line for the domain of $f$.

   2. Mark each critical number on the number line.

   3. Choose a test value in each subinterval created by the critical numbers.

   4. Calculate $f'(c)$ for each test value. If $f'(c) > 0$, $f$ is increasing on that subinterval; if $f'(c) < 0$, $f$ is decreasing.

Example 1: $f(x) = x^3 - 2x^2 + 7$

• Find $f'(x) = 3x^2 - 4x$

• Set $f'(x) = 0$: $3x^2 - 4x = 0 \implies x(3x - 4) = 0 \implies x = 0, \frac{4}{3}$

• Test intervals: $(-\infty, 0)$, $(0, \frac{4}{3})$, $(\frac{4}{3}, \infty)$

• Choose test points and evaluate $f'(x)$ to determine sign and thus intervals of increase/decrease.

Example 2: $f(x) = x^4 + 6x^2 + 9x - 1$

• Find $f'(x) = 4x^3 + 12x + 9$

• Set $f'(x) = 0$ and solve for $x$ (may require factoring or numerical methods).

• Test intervals between critical numbers as above.

Analyzing the Graph of the Derivative

Sometimes, you are given the graph of $f'$ (the derivative) and asked about the behavior of $f$:

• Critical values: Occur where $f'(x) = 0$ (crosses the x-axis) or is undefined.

• Increasing intervals: Where $f'(x) > 0$ (above the x-axis).

• Decreasing intervals: Where $f'(x) < 0$ (below the x-axis).

Summary Table: Increasing, Decreasing, and Critical Points

Interval	Sign of $f'(x)$	Behavior of $f(x)$
$(a, b)$	$f'(x) > 0$	Increasing
$(a, b)$	$f'(x) < 0$	Decreasing
At $x = c$	$f'(c) = 0$ or DNE	Critical Point

Additional info: In business calculus, identifying intervals of increase and decrease helps in maximizing profit, minimizing cost, and understanding trends in economic models.