Question

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. $$f(x)=|x|, \quad-1<x<2$$

Name: Problem 15, Chapter 4: University Calculus: Early Transcendentals
Uploaded: 2020-06-28T05:09:59-08:00
Duration: 56 s
Channel: Katelyn Chen
Description: 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

   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.
$$f(x)=|x|, \quad-1<x<2$$

University Calculus: Early Transcendentals

Joel Hass, Maurice… 3rd Edition

Chapter 4, Problem 15

Instant Answer

Solved by Expert Katelyn Chen

06/28/2020

Step 1

The function given is $ f(x) = |x| $, which is the absolute value function. The domain specified is $-1 < x < 2$, which means we are considering the function only for values of $ x $ strictly between $-1$ and $2$. Show more…

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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. $$f(x)=|x|, \quad-1<x<2$$

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Key Concepts

Absolute Extreme Values

Absolute extreme values refer to the overall maximum and minimum values of a function when considering its entire domain. For a function f, an absolute minimum is a value f(c) such that f(c) ? f(x) for all x in the domain, and an absolute maximum is a value f(d) such that f(d) ? f(x) for all x. In contexts with an open domain, a function may fail to attain an absolute maximum or minimum if such values would occur at points excluded from the domain.

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed and bounded interval, then it is guaranteed to have both an absolute maximum and an absolute minimum on that interval. This theorem highlights the importance of closed domains because endpoints are included. In cases where the domain is open or not closed, as in the given example, the function may not achieve one or both of these extremes even if it is continuous throughout its domain.

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