5.2 Extrema On An Interval Homework

5.2 Extrema on an Interval: A Comprehensive Guide to Homework Problems

Finding extrema (maximum and minimum values) of a function on a given interval is a crucial concept in calculus. This guide delves into the intricacies of Section 5.2, focusing on solving various homework problems related to extrema on an interval. We'll explore different techniques, address common challenges, and provide a structured approach to tackle these problems effectively.

Understanding Extrema

Before diving into problem-solving, let's solidify our understanding of extrema. An extremum is a point where a function achieves a maximum or minimum value. We differentiate between:

• Absolute (Global) Extrema: The highest or lowest value of the function across the entire interval. There can only be one absolute maximum and one absolute minimum.
• Local (Relative) Extrema: The highest or lowest value of the function within a small neighborhood of a point. A function can have multiple local extrema.

Critical Points and Endpoint Analysis: The Core Strategy

The key to finding extrema on a closed interval [a, b] involves a two-pronged approach:

1. Finding Critical Points: These are points within the interval where the derivative is zero or undefined. Critical points are potential candidates for local extrema. To find them:

   • Calculate the first derivative, f'(x).
   • Solve the equation f'(x) = 0 to find points where the derivative is zero.
   • Identify points where the derivative is undefined (e.g., points where the function is not differentiable).

2. Endpoint Analysis: Evaluate the function at the endpoints of the interval, f(a) and f(b). Endpoints can often be absolute extrema even if they are not critical points.

The Procedure:

1. Find the derivative: Calculate f'(x).
2. Find critical points: Solve f'(x) = 0 and identify points where f'(x) is undefined within the interval [a, b].
3. Evaluate the function: Compute f(x) at all critical points found in step 2 and at the endpoints a and b.
4. Compare the values: The largest value is the absolute maximum, and the smallest value is the absolute minimum on the interval [a, b].

Example Problems and Solutions

Let's work through several example problems to illustrate the process:

Problem 1: Find the absolute extrema of f(x) = x³ - 6x² + 9x + 2 on the interval [0, 4].

Solution:

1. Find the derivative: f'(x) = 3x² - 12x + 9.
2. Find critical points: Set f'(x) = 0: 3x² - 12x + 9 = 0. This simplifies to x² - 4x + 3 = 0, which factors as (x - 1)(x - 3) = 0. Thus, the critical points are x = 1 and x = 3. Both are within the interval [0, 4].
3. Evaluate the function:
   • f(0) = 2
   • f(1) = 6
   • f(3) = 2
   • f(4) = 6
4. Compare the values: The absolute maximum is 6, which occurs at x = 1 and x = 4. The absolute minimum is 2, which occurs at x = 0 and x = 3.

Problem 2: Find the absolute extrema of g(x) = x⁴ - 8x² + 16 on the interval [-3, 2].

Solution:

1. Find the derivative: g'(x) = 4x³ - 16x.
2. Find critical points: Set g'(x) = 0: 4x³ - 16x = 0. This factors as 4x(x² - 4) = 0, giving critical points x = 0, x = 2, and x = -2. All are within the interval [-3, 2].
3. Evaluate the function:
   • g(-3) = 25
   • g(-2) = 0
   • g(0) = 16
   • g(2) = 0
4. Compare the values: The absolute maximum is 25 at x = -3. The absolute minimum is 0 at x = -2 and x = 2.

Problem 3: Find the absolute extrema of h(x) = √(x) on the interval [0, 4].

Solution:

1. Find the derivative: h'(x) = 1/(2√x).
2. Find critical points: h'(x) is undefined at x = 0. There are no values of x where h'(x) = 0.
3. Evaluate the function:
   • h(0) = 0
   • h(4) = 2
4. Compare the values: The absolute maximum is 2 at x = 4. The absolute minimum is 0 at x = 0. Notice that the minimum occurs where the derivative is undefined.

Handling Functions with Undefined Derivatives

Some functions have derivatives that are undefined at certain points within the interval. These points are still potential candidates for extrema. Consider the following example:

Problem 4: Find the absolute extrema of k(x) = |x| on the interval [-2, 2].

Solution:

The derivative of k(x) = |x| is undefined at x = 0. However, we can still analyze this point.

1. Derivative: k'(x) = 1 for x > 0 and k'(x) = -1 for x < 0; undefined at x = 0.
2. Critical points: x = 0 (where the derivative is undefined).
3. Evaluate the function:
   • k(-2) = 2
   • k(0) = 0
   • k(2) = 2
4. Compare values: The absolute maximum is 2 at x = -2 and x = 2. The absolute minimum is 0 at x = 0.

Extending the Concepts: Closed vs. Open Intervals

The problems discussed so far have involved closed intervals [a, b]. When dealing with open intervals (a, b) or half-open intervals, the analysis is slightly different. Absolute extrema might not exist on open intervals. You'll need to consider the limits of the function as x approaches a and b. If the limits are finite and represent the smallest or largest values, they can be considered absolute minima or maxima, respectively.

Advanced Techniques and Considerations

For more complex functions, numerical methods or graphing calculators might be necessary to approximate critical points and evaluate the function. Understanding the behavior of the function (increasing/decreasing intervals, concavity) can aid in visualizing and interpreting the results. Remember that the Closed Interval Method provides a powerful and systematic approach to find absolute extrema on closed intervals.

Practice Problems

To solidify your understanding, try solving these problems:

1. Find the absolute extrema of f(x) = x⁴ - 4x³ + 4x² on the interval [-1, 3].
2. Find the absolute extrema of g(x) = x²e⁻ˣ on the interval [0, 5].
3. Find the absolute extrema of h(x) = 1/x on the interval (0, 1). (Consider limits)
4. Find the absolute extrema of k(x) = sin(x) + cos(x) on the interval [0, π].
5. Find the absolute extrema of m(x) = x^(2/3) on the interval [-1, 1].

By diligently working through these examples and practice problems, you will develop a solid understanding of how to find extrema on an interval and successfully tackle similar homework assignments in Section 5.2 of your calculus textbook. Remember to always clearly define your steps and justify your conclusions. Good luck!