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Chapter 12: Problem 62

If one endpoint is a relative maximum, must the other be a relative minimum? Explain your answer.

Short Answer

Expert verified
If one endpoint of a function is a relative maximum, it does not necessarily imply that the other endpoint must be a relative minimum. The behavior and shape of the function determine the existence of a relative minimum or not, as illustrated in the counterexample with the function \(f(x) = x^3\) on the interval \([-2, 2]\).

Step by step solution

01

Definition of Relative Maximum and Minimum

A relative maximum of a function is a point where the function has a local high point, meaning its value is greater than its neighboring values. A relative minimum of a function is a point where the function has a local low point, meaning its value is less than its neighboring values.
02

Analyze the Endpoints

Endpoints of a function are values at the end of the domain of a function for which the function is defined. These points must be considered to determine the existence of relative maximum or minimum.
03

Counterexample

Consider the function \(f(x) = x^3\) on the interval \([-2, 2]\). The graph of this function looks like a "hill" that slopes upward as you move from left to right. The endpoint at \(x=-2\) is a relative maximum, because the function's value is greater than its neighbors. However, the endpoint at \(x=2\) is not a relative minimum, as the function's value is greater than its neighbors.
04

Conclusion

If one endpoint of a function is a relative maximum, it does not necessarily mean that the other endpoint must be a relative minimum. This can be seen in the counterexample provided. The function's behavior and shape determine whether its other endpoint is a relative minimum or not.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Relative Maximum
In calculus, identifying a relative maximum is key to understanding the behavior of a function in a specific interval. A relative maximum occurs at a point where the function reaches a peak higher than any nearby points. Imagine a small "hill" on a graph. This is where the top of the hill signifies a relative maximum point.

At this point, the derivative of the function changes from positive to negative, indicating a switch from increasing to decreasing behavior. It's a local peak rather than the highest point over the entire function, and it helps us see the smaller patterns within broader trends.
  • The presence of a relative maximum doesn't imply anything specific about the other points or endpoints of the function.
  • Understanding where the slope changes can aid in graphing and predicting function behavior.
Relative Minimum
Similar to a relative maximum, a relative minimum is a point on a function where it reaches a local trough or bottom. It's akin to the bottom of a small "valley" on a graph, where the function value is smaller than all those immediately surrounding it.

At a relative minimum, the function's derivative changes from negative to positive. This switch displays a reversal from a downward trend to an upward one. It occurs in localized regions rather than across the entire domain of the function.
  • Though a function might have several relative minima, none are guaranteed to be at the endpoints.
  • These points help identify areas where the function reaches a temporary low within a specific section.
Endpoints of Function
Endpoints are the points at the boundary of a function's domain. They often need special attention to determine whether they are relative maxima or minima.

The rule for endpoints is that they aren't strictly bound to be either relative maximum or minimum, but they are still integral in analyzing a function's overall behavior. Evaluating endpoints requires understanding their context in the entire domain.
  • Endpoints may or may not exhibit extrema because evaluating them relies heavily on the function's behavior as a whole.
  • A function like \(f(x) = x^3\) on the interval \([-2, 2]\) demonstrates that endpoints don't always toggle between being maxima or minima.
This shows that while endpoints factor into a function's shape and behavior, they don't inherently dictate the presence of relative extrema at the other end.

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Most popular questions from this chapter

Sketch the graph of the given function, indicating (a) \(x\) - and \(y\) -intercepts, (b) extrema, (c) points of inflection, \((d)\) behavior near points where the function is not defined, and (e) behavior at infinity. Where indicated, technology should be used to approximate the intercepts, coordinates of extrema, and/or points of inflection to one decimal place. Check your sketch using technology. \(f(x)=x^{2}+2 x+1\)

A time- series study of the demand for higher education, using tuition charges as a price variable, yields the following result: $$\frac{d q}{d p} \cdot \frac{p}{q}=-0.4$$ where \(p\) is tuition and \(q\) is the quantity of higher education. \(\mathbf{2 5}\). Which of the following is suggested by the result? (A) As tuition rises, students want to buy a greater quantity \(\quad \mathbf{2 6}\). of education. (B) As a determinant of the demand for higher education, income is more important than price. (C) If colleges lowered tuition slightly, their total tuition receipts would increase. (D) If colleges raised tuition slightly, their total tuition receipts would increase. (E) Colleges cannot increase enrollments by offering larger scholarships.

Daily oil production in Mexico and daily U.S. oil imports from Mexico during \(2005-2009\) can be approximated by $$\begin{array}{ll}P(t)=3.9-0.10 t \text { million barrels } & (5 \leq t \leq 9) \\ I(t)=2.1-0.11 t \text { million barrels } & (5 \leq t \leq 9) \end{array}$$ where \(t\) is time in years since the start of \(2000 .^{41}\) Graph the function \(I(t) / P(t)\) and its derivative. Is the graph of \(I(t) / P(t)\) concave up or concave down? The concavity of \(I(t) / P(t)\) tells you that: (A) The percentage of oil produced in Mexico that was exported to the United States was decreasing. (B) The percentage of oil produced in Mexico that was not exported to the United States was increasing. (C) The percentage of oil produced in Mexico that was exported to the United States was decreasing at a slower rate. (D) The percentage of oil produced in Mexico that was exported to the United States was decreasing at a faster rate.

Rewrite the statements and questions in mathematical notation. The population \(P\) is currently 10,000 and growing at a rate of 1,000 per year.

Your friend tells you that he has found a continuous function defined on \((-\infty,+\infty)\) with exactly two critical points, each of which is a relative maximum. Can he be right?
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