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.f(x) = |x|, −1 \u003c x \u003c 2

Accepted Answer

Step 1: Begin by understanding the function f(x) = |x|, which represents the absolute value function. The graph of f(x) = |x| is a V-shaped graph with its vertex at the origin (0,0). However, we need to consider the domain given: -1 \u003c x \u003c 2. Step 2: Sketch the graph of f(x) = |x| within the specified domain. Since the domain is -1 \u003c x \u003c 2, the graph will start just to the right of x = -1 and end just before x = 2. The vertex of the graph at (0,0) will be included in this domain. Step 3: Identify the behavior of the graph within the domain. As x approaches -1 from the right, the function value approaches 1, and as x approaches 2 from the left, the function value approaches 2. The graph is symmetric about the y-axis, and the lowest point within the domain is at the vertex (0,0) where f(x) = 0. Step 4: Determine the absolute extrema. The absolute minimum value of f(x) within the domain -1 \u003c x \u003c 2 is at x = 0, where f(x) = 0. There is no absolute maximum value because the function value increases indefinitely as x approaches 2 from the left. Step 5: Relate your findings to Theorem 1, which states that a continuous function on a closed interval [a, b] has both an absolute maximum and minimum. However, the interval -1 \u003c x \u003c 2 is not closed, so the theorem does not apply directly. The function has an absolute minimum at x = 0 but no absolute maximum within the open interval.

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.

f(x) = |x|, −1 < x < 2

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Absolute Extrema

Graphing 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.f(x) = |x|, −1 < x < 2