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Chapter 2: Problem 114

Determine whether each statement makes sense or does not make sense, and explain your reasoning. My graph is decreasing on \((-\infty, a)\) and increasing on \((a, \infty)\) so \(f(a)\) must be a relative maximum.

Short Answer

Expert verified
Yes, the statement does make sense. If a function is decreasing to the left of a point and increasing to the right, that point is indeed a relative maximum.

Step by step solution

01

Identifying the Given Information

The statement given is: 'My graph is decreasing on \((-\infty, a)\) and increasing on \((a, \infty)\) so \(f(a)\) must be a relative maximum.' This tells us that the function \(f(x)\) decreases as \(x\) approaches \(a\) from the left (that is, over the interval \((-\infty, a)\)), and increases as \(x\) moves away from \(a\) on the right (over the interval \((a, \infty)\)). The claim is that \(f(a)\) must be a relative maximum.
02

Understanding Relative Maximum

A point \(f(a)\) is a relative (or local) maximum if it is greater than (or equal to) the values of \(f(x)\) for x near a. This means that for a small range of x-values around \(a\), \(f(a)\) is the largest value.
03

Correlating the Given Information and Concept of Relative Maximum

If the function is decreasing on \((-\infty, a)\), that means \(f(a)\) is greater than or equal to any point to its left. Similarly, if the function is increasing on \((a, \infty)\), \(f(a)\) is also greater or equal to any point to its right over this interval. Thus, \(f(a)\) does actually meet the definition of a relative maximum.

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