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Chapter 2: Problem 104

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(f(x)=-x^{3}+4 x,\) then the graph of \(f\) falls to the left and falls to the right.

Short Answer

Expert verified
The statement is false. If \(f(x)=-x^{3}+4x\), then the graph of \(f\) should rise to the left and fall to the right. So the statement should be corrected to 'If \(f(x)=-x^{3}+4x\), then the graph of \(f\) rises to the left and falls to the right.'

Step by step solution

01

Analyze the function

First, let's look at the function \(f(x)=-x^{3}+4x\). The degree of the function is 3 (an odd number) and the leading coefficient is -1 (a negative number). According to the rule stated in the analysis, for an odd-degree polynomial with a negative leading coefficient, the function must rise to the left and fall to the right.
02

Determine the truthfulness of the statement

The statement suggests that the function \(f(x)=-x^{3}+4x\) falls to the left and falls to the right. This is inconsistent with our analysis in step 1, where we found that the function should rise to the left and fall to the right. Therefore, the statement is false.
03

Correct the statement

In order to render the statement true, it needs to correctly describe the end behavior of the function \(f(x)=-x^{3}+4x\). Changing it to: 'If \(f(x)=-x^{3}+4x\), then the graph of \(f\) rises to the left and falls to the right.' This now accurately reflects the end behavior of the function.

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