Analyzing Functions: Intervals, Extrema, and Concavity

Analyzing Functions: Intervals, Extrema, and Concavity

Introduction

Understanding the behavior of functions is a fundamental skill in Business Calculus. This includes identifying where a function is increasing or decreasing, locating relative maxima and minima, and determining intervals of concavity and inflection points. These concepts are essential for optimization and economic modeling.

Key Concepts in Function Analysis

Intervals of Increase and Decrease

A function f(x) is said to be increasing on an interval if, as x increases, f(x) also increases. Conversely, it is decreasing if f(x) decreases as x increases.

• Increasing Interval: Where the graph rises as you move left to right.

• Decreasing Interval: Where the graph falls as you move left to right.

Mathematical Definition:

• Increasing: $f'(x) > 0$ on the interval

• Decreasing: $f'(x) < 0$ on the interval

Relative (Local) Extrema

Relative maxima and minima are points where the function reaches a highest or lowest value locally.

• Relative Maximum: A point where $f(x)$ changes from increasing to decreasing ($f'(x)$ changes from positive to negative).

• Relative Minimum: A point where $f(x)$ changes from decreasing to increasing ($f'(x)$ changes from negative to positive).

First Derivative Test: Used to identify relative extrema by analyzing sign changes in $f'(x)$.

Concavity and Inflection Points

Concavity describes the direction the graph bends:

• Concave Up: The graph bends upwards like a cup ($f''(x) > 0$).

• Concave Down: The graph bends downwards like a frown ($f''(x) < 0$).

Inflection Point: A point where the graph changes concavity (from up to down or vice versa). This occurs where $f''(x) = 0$ and the sign of $f''(x)$ changes.

Example: Analyzing a Function Graph

Given a graph of $f(x)$, answer the following for each function:

• f(x) increasing intervals: Identify intervals where the graph rises.

• f(x) decreasing intervals: Identify intervals where the graph falls.

• f(x) relative max: Points where the graph peaks locally.

• f(x) relative min: Points where the graph dips locally.

• f(x) concave up: Intervals where the graph is shaped like a cup.

• f(x) concave down: Intervals where the graph is shaped like a frown.

• f(x) inflection: Points where the graph changes concavity.

Example Table: Summary of Function Behavior

Property	Description	How to Identify
Increasing Interval	Where $f(x)$ rises	$f'(x) > 0$
Decreasing Interval	Where $f(x)$ falls	$f'(x) < 0$
Relative Maximum	Local highest point	$f'(x)$ changes + to -
Relative Minimum	Local lowest point	$f'(x)$ changes - to +
Concave Up	Graph bends upward	$f''(x) > 0$
Concave Down	Graph bends downward	$f''(x) < 0$
Inflection Point	Concavity changes	$f''(x) = 0$ and sign changes

Applications in Business Calculus

• Optimization: Finding maximum profit or minimum cost by locating relative extrema.

• Economic Modeling: Understanding how revenue, cost, or demand functions behave over time.

Summary

Analyzing the intervals of increase/decrease, relative extrema, and concavity of functions is crucial for solving real-world business problems. Mastery of these concepts allows for effective optimization and interpretation of economic models.