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Chapter 2: 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
\(\)f(a)\ is a relative minimum

Step by step solution

01

Understand the conditions for a relative maximum

A relative maximum of a function is a point where the value of the function at this point is greater than or equal to the values of the function at all nearby points. According to the first derivative test, if the function changes its status from decreasing to increasing at a certain point, that point is a relative minimum. In other words, if the derivative of the function changes its sign from negative to positive, then that point is a relative minimum.
02

Analyze the given statement

In the given statement, the graph is decreasing on the interval \((- \infty, a)\) and increasing on the interval \((a, \infty)\). This indicates that the derivative of the function is negative up to 'a' and is positive thereafter. That means the function moves from a decreasing state to an increasing state at 'a', which indicates that 'a' is a relative minimum point.
03

Conclusion

Therefore, the given statement 'My graph is decreasing on \((- \infty, a)\) and increasing on \((a, \infty)\) so \(f(a)\) must be a relative maximum' does not make sense. Instead, based on the given conditions, \(f(a)\) would be a relative minimum, not a relative maximum.

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