Question

Finding Extrema from GraphsIn Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.g(x) = {−x, 0 ≤ x \u003c 1x − 1, 1 ≤ x ≤ 2

Accepted Answer

Step 1: Begin by understanding the function g(x) which is piecewise defined. The function is given as g(x) = -x for 0 ≤ x \u003c 1 and g(x) = x - 1 for 1 ≤ x ≤ 2. This means the function has two different expressions depending on the interval of x. Step 2: Sketch the graph of the function g(x). For the interval 0 ≤ x \u003c 1, plot the line g(x) = -x, which is a straight line with a negative slope passing through the origin. For the interval 1 ≤ x ≤ 2, plot the line g(x) = x - 1, which is a straight line with a positive slope starting at the point (1, 0). Step 3: Identify the endpoints and any points of interest. The function is defined on the interval [0, 2]. Check the values of g(x) at the endpoints: g(0) = 0 and g(2) = 1. Also, check the value at x = 1, where the function transitions from one piece to another: g(1) = 0. Step 4: Determine the absolute extrema by comparing the values of g(x) at the endpoints and any critical points. Since g(0) = 0, g(1) = 0, and g(2) = 1, the absolute minimum value is 0, occurring at both x = 0 and x = 1. The absolute maximum value is 1, occurring at x = 2. Step 5: Explain how the answer is consistent with Theorem 1, which states that if a function is continuous on a closed interval, it must have both a maximum and minimum value on that interval. Although g(x) is not continuous at x = 1, it is continuous on the intervals [0, 1) and [1, 2], allowing us to find the extrema within the domain.

Finding Extrema from Graphs

In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.

g(x) = {−x, 0 ≤ x < 1
x − 1, 1 ≤ x ≤ 2

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Concetti chiave

Absolute Extrema

Piecewise Functions

Theorem 1 (Extreme Value Theorem)

Finding Extrema from GraphsIn Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.g(x) = {−x, 0 ≤ x < 1x − 1, 1 ≤ x ≤ 2