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Chapter 1: Problem 133

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
No, the statement does not make sense because the function value at \(f(a)\) should be a relative minimum, not a relative maximum, based on the behaviour of the function.

Step by step solution

01

Dissect the Statement

Let's read and dissect each part of the sentence separately. The sentence makes a claim about a function that it decreases on interval \((-\infty, a)\) and increases on interval \((a, \infty)\). Because of this behaviour of the function, the statement claims that the function value at \(a\), denoted as \(f(a)\), is a relative maximum.
02

Define Increasing and Decreasing Intervals of a Function

The function is said to be decreasing on an interval if for every pair of numbers within that interval, if one number is larger, then its function value is smaller. Similarly, a function is said to be increasing on an interval if for every pair of numbers within that interval, if one number is larger, then its function value is also larger. Here, the statement says that the function is decreasing as we approach \(a\) from negative infinity, and then it starts increasing as soon as we move past \(a\) towards positive infinity.
03

Define Relative Maximum

A point is at a relative maximum if the function values at that point are greater than or equal to all function values in its immediate vicinity. In other words, in a small enough open interval around that point, it has the largest function value.
04

Interpreting the Given Statement

Given that \(f\) is decreasing until \(a\) and then starts increasing, at point \(a\), it should be the smallest point (relative minimum) in its immediate vicinity. Therefore, the mystery statement does not make sense because clearly, it contradicts the definition of a relative maximum.

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