Skip to main content
Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.1.65
Chapter 4, Problem 4.1.65

Theory and Examples


[Technology Exercise] Graph the functions in Exercises 63–66. Then find the extreme values of the function on the interval and say where they occur.


h(x) = |x + 2| − |x − 3|, −∞ < x < ∞

Verified step by step guidance
1
Step 1: Understand the function h(x) = |x + 2| − |x − 3|. This function involves absolute values, which can be split into piecewise functions based on the critical points where the expressions inside the absolute values change sign.
Step 2: Identify the critical points for the absolute value expressions. The critical points are x = -2 and x = 3, where the expressions inside the absolute values change sign.
Step 3: Break down the function into piecewise components based on the critical points. For x < -2, both expressions are negative, for -2 ≤ x < 3, the first expression is non-negative and the second is negative, and for x ≥ 3, both expressions are non-negative.
Step 4: Graph the piecewise function by evaluating h(x) in each interval: x < -2, -2 ≤ x < 3, and x ≥ 3. This will help visualize the behavior of the function across the entire domain.
Step 5: Determine the extreme values by analyzing the graph and evaluating the function at the critical points and endpoints of the intervals. The extreme values occur where the function reaches its maximum or minimum values within the given domain.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Absolute Value Functions

Absolute value functions, such as |x + 2| and |x - 3|, measure the distance of a number from zero on the number line, always yielding a non-negative result. These functions create V-shaped graphs and are crucial for understanding how the function h(x) = |x + 2| − |x − 3| behaves, especially at points where the expressions inside the absolute values change sign.
Recommended video:
06:37
Average Value of a Function

Graphing Piecewise Functions

Graphing piecewise functions involves plotting different expressions over specified intervals. For h(x) = |x + 2| − |x − 3|, the function can be broken into segments based on the critical points where x + 2 = 0 and x - 3 = 0. Understanding how to graph these segments helps visualize the function's behavior and identify extreme values.
Recommended video:
05:36
Piecewise Functions

Finding Extreme Values

Extreme values of a function are its maximum and minimum values within a given interval. To find these for h(x), analyze the critical points where the derivative is zero or undefined, and evaluate the function at these points and endpoints of the interval. This process helps determine where the function reaches its highest or lowest values.
Recommended video:
05:12
Finding Global Extrema (Extreme Value Theorem)
Related Practice
Textbook Question

101. In Exercises 101 and 102, the graph of f' is given. Determine x-values corresponding to local minima, local maxima, and inflection points for the graph of f.

Textbook Question

99. In Exercises 99 and 100, the graph of f' is given. Determine x-values corresponding to inflection points for the graph of f.

Textbook Question

Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.

7. y=sin|x|, -2π≤x≤2π

Textbook Question

Find values of a and b such that the function


ƒ(𝓍) = (a𝓍 + b) / 𝓍² ―1)


has a local extreme value of 1 at 𝓍 = 3. Is this extreme value a local maximum or a local minimum? Give reasons for your answer.

Textbook Question

Initial Value Problems


Solve the initial value problems in Exercises 71–90.


dy/dx = 3x⁻²ᐟ³, y(−1) = −5

Textbook Question

Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.

4. y=9/14x^(1/3)(x^2-7)