/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 100 Let \(f(x)=\frac{|x|}{x} .\) The... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 100

Let \(f(x)=\frac{|x|}{x} .\) Then \(f(-2)=-1\) and \(f(2)=1 .\) Therefore \(f(-2)<0

Short Answer

Expert verified
Explain your answer. Answer: No, the fact that there is no value c for which \(f(c)=0\) does not violate the Intermediate Value Theorem. This is because the function \(f(x)=\frac{|x|}{x}\) is not continuous on the interval from -2 to 2 due to its undefined value at x=0. The Intermediate Value Theorem can only be applied if the function is continuous on the considered interval.

Step by step solution

01

Examine the function and its continuity

Let's examine the given function \(f(x)=\frac{|x|}{x}\). We know that the absolute value is continuous everywhere; however, the function is undefined when x=0 because division by 0 is not allowed. Thus, the given function is not continuous on the interval from -2 to 2.
02

Consider the Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on an interval [a, b], and a value k lies between the function's values at a and b (i.e., \(f(a)k>f(b)\)), then there exists a point 'c' in the interval where the function takes that intermediate value (i.e., \(f(c)=k\)). In this problem, we have \(f(-2)=-1\), \(f(2)=1\), and we're looking for a value \(c\) between -2 and 2 where \(f(c)=0\). However, since the function is not continuous on the interval, we cannot directly apply the Intermediate Value Theorem.
03

Conclusion

The fact that there is no value \(c\) between -2 and 2 for which \(f(c)=0\) does not violate the Intermediate Value Theorem because the function \(f(x)=\frac{|x|}{x}\) is not continuous on the interval from -2 to 2. The Intermediate Value Theorem can only be applied if the function is continuous on the considered interval. So in this case, the Intermediate Value Theorem does not apply.

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