Critical Points and Absolute Extrema in Calculus

Critical Points and Absolute Extrema

Introduction

This section explores how to find and classify the extreme values (maximum and minimum) of functions, both locally and absolutely, using calculus. The concepts of critical points, local extrema, and absolute extrema are fundamental for analyzing the behavior of functions on given intervals.

Local Extrema

Local extrema refer to the highest or lowest points (peaks or valleys) of a function within a certain neighborhood. These are also called local maximum and local minimum values.

• Definition: A function has a local extremum at x = c when $f'(c) = 0$.

• At these points, the slope of the tangent line to the graph is zero.

• Local extrema can occur at critical points or endpoints of the interval.

Critical Points

Critical points are values of x in the domain of a function where the derivative is zero or undefined.

• Definition: If $f'(x) = 0$ or $f'(x)$ is undefined, then x is a critical point.

• Critical points are candidates for local maxima, minima, or points of inflection.

Example: Finding Critical Points

• Given $f(x) = 6x^2 - x^3$, find all critical points.

• Compute the derivative: $f'(x) = 12x - 3x^2$.

• Set $f'(x) = 0$: $12x - 3x^2 = 0 \Rightarrow 3x(4 - x) = 0 \Rightarrow x = 0$ or $x = 4$.

• Thus, the critical points are at $x = 0$ and $x = 4$.

Absolute Extrema

Absolute extrema are the highest and lowest values of a function on a closed interval, including endpoints. These are also called absolute maximum and absolute minimum values.

• Absolute extrema can occur at critical points or at the endpoints of the interval.

• To find absolute extrema, evaluate the function at all critical points and endpoints, then compare the values.

Five-Step Process for Finding Absolute Extrema

1. Find all critical points of the function in the closed interval.

2. Find the y-values for all critical points (plug critical points into the original function).

3. Find the y-values at the endpoints (plug endpoint values into the original function).

4. The largest y-value is the absolute maximum in the closed interval.

5. The smallest y-value is the absolute minimum in the closed interval.

Example: Absolute Extrema on a Closed Interval

• Given $y = 4 - x^2$ on the interval $-2 \leq x \leq 1$, find the absolute maximum and minimum.

• Compute the derivative: $y' = -2x$.

• Set $y' = 0$: $-2x = 0 \Rightarrow x = 0$ (critical point).

• Evaluate $y$ at $x = -2, 0, 1$:

  • $y(-2) = 4 - (-2)^2 = 0$

  • $y(0) = 4 - 0 = 4$

  • $y(1) = 4 - 1 = 3$

• Absolute maximum: $y = 4$ at $x = 0$

• Absolute minimum: $y = 0$ at $x = -2$

Worked Examples

• Example 1: Given a graph, locate the minimum and maximum values over specified intervals by identifying the lowest and highest points on the graph within those intervals.

• Example 4: Find all critical points of $f(x) = \frac{x^2}{x^2 - 2}$.

  • Compute the derivative using the quotient rule:

  • $f'(x) = \frac{(2x)(x^2 - 2) - x^2(2x)}{(x^2 - 2)^2} = \frac{2x(x^2 - 2) - 2x^3}{(x^2 - 2)^2}$

  • Simplify and set $f'(x) = 0$ to find critical points.

• Example 7: Find the absolute maximum and minimum of $y = \csc(\theta)$ on the interval $-\frac{\pi}{5} \leq \theta \leq \frac{\pi}{4}$.

  • Evaluate $y$ at endpoints and any critical points within the interval.

Summary Table: Steps for Finding Absolute Extrema

Step	Description
1	Find all critical points in the closed interval
2	Evaluate the function at each critical point
3	Evaluate the function at the endpoints
4	Compare all values to determine the absolute maximum and minimum

Key Terms

• Critical Point: A point where $f'(x) = 0$ or $f'(x)$ is undefined.

• Local Maximum/Minimum: The highest/lowest value of a function in a neighborhood.

• Absolute Maximum/Minimum: The highest/lowest value of a function on a closed interval.

Applications

• Optimization problems in physics, engineering, and economics often require finding absolute extrema.

• Critical points help identify where a function changes direction or has a peak/valley.

Homework Practice

• Section 4.1: Problems 21-33 odd and 37-51 odd