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Chapter 12: Problem 45

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?

Short Answer

Expert verified
No, your friend can't be right. It is impossible for a continuous function with exactly two critical points to have each of those points be a relative maximum. This is because the second derivative must change sign from negative to positive, passing through zero at some point in between the two maxima, contradicting the initial condition of having only two critical points.

Step by step solution

01

Understand critical points and relative maxima

A critical point of a function is a point where the function's first derivative is either equal to zero or not defined. A relative maximum is a point where the function's value is greater than the values of the function at neighboring points. If a point is a relative maximum, its first derivative must be equal to zero.
02

Analyze conditions for the continuous function

As the function is continuous and if it has exactly two critical points, each of which is a relative maximum, we can analyze this by considering the concavity of the function. The second derivative test is used to determine whether a critical point is a relative maximum or minimum: 1. If the second derivative is positive at the critical point, the point is a relative minimum. 2. If the second derivative is negative at the critical point, the point is a relative maximum. 3. If the second derivative is equal to zero or not defined at the critical point, the test is inconclusive.
03

Check if the given function can satisfy the conditions

Let's suppose that our two critical points are \(a\) and \(b\) such that \(a<b\). We know that at both points, the first derivative is equal to zero. Also, as per the given conditions, both \(a\) and \(b\) are relative maxima; hence, the second derivative must be negative at both points. Now, consider the interval \((a, b)\). As both critical points are relative maxima, the second derivative must change sign from negative to positive, passing through zero at some point \(c\) in the interval \((a, b)\). This means that there is another critical point \(c\) within this interval. But this contradicts the initial condition that there are exactly two critical points. Therefore, finding a continuous function with exactly two critical points, each of which being a relative maximum, is not possible.
04

Conclusion

No, your friend can't be right. It is impossible for a continuous function with exactly two critical points to have each of those points be a relative maximum.

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Key Concepts

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

Continuous Function
A continuous function is one that does not have any abrupt changes in value, meaning that for every point within its domain, the function can be drawn without lifting the pencil from the paper. Formally, a function is continuous at a point if the limit of the function as it approaches the point is equal to the function's value at that point. This concept is foundational in calculus because it allows for the analysis and understanding of functions over intervals.

Continuous functions have important properties that affect their behavior. For example, according to the Intermediate Value Theorem, if a function is continuous on a closed interval and takes different values at the endpoints of the interval, then it must take any value between those endpoints at some point within the interval. This property is fundamental for understanding the behavior of functions as they relate to critical points and extremum values, such as relative maxima.
Relative Maximum
A relative maximum of a function is a point where the function's value is higher than the value at any nearby points. In other words, it's a local high point on the graph of the function. When a function reaches a relative maximum, the slope of the tangent to the function (represented by the first derivative) is zero. This is because at the very top of a hill, there is no incline鈥攎eaning the slope is flat, or zero.

To determine if a critical point is a relative maximum, one can use the First Derivative Test: observe the sign of the first derivative before and after the critical point. If it changes from positive to negative, the function is increasing before the point and decreasing after, indicating a relative maximum at the critical point. Detailed understanding of this concept is imperative for analyzing the behavior of functions and for substantiating the impossibility of a continuous function having exactly two critical points with both being relative maxima.
Second Derivative Test
The second derivative test is a useful criterion in calculus for determining whether a given critical point is a relative maximum, minimum, or a saddle point. This test involves taking the second derivative of a function, which tells us about the concavity of the function at a point.

Here's the gist of the second derivative test:

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Most popular questions from this chapter

The weekly sales of Honolulu Red Oranges is given by \(q=1,000-20 p\). Calculate the price elasticity of demand when the price is \(\$ 30\) per orange (yes, \(\$ 30\) per orange \(^{63}\) ). Interpret your answer. Also, calculate the price that gives a maximum weekly revenue, and find this maximum revenue. HINT [See Example 1.]

If two quantities \(x\) and \(y\) are related by a linear equation, how are their rates of change related?

Your company is the largest sock manufacturer in the solar system, and production is automated through the use of androids and robots. In order to meet production deadlines, your company calculates that the numbers of androids and robots must satisfy the constraint $$x y=1,000,000$$ where \(x\) is the number of androids and \(y\) is the number of robots. Your company currently uses 5000 androids and is increasing android deployment at a rate of 200 per month. How fast is it scrapping robots? HINT [See Example 4.]

Your other friend tells you that she has found a continuous function with two critical points, one a relative minimum and one a relative maximum, and no point of inflection between them. Can she be right?

A circular conical vessel is being filled with ink at a rate of \(10 \mathrm{~cm}^{3} / \mathrm{s} .\) How fast is the level rising after \(20 \mathrm{~cm}^{3}\) have been poured in? The cone has height \(50 \mathrm{~cm}\) and radius \(20 \mathrm{~cm}\) at its brim. (The volume of a cone of height \(h\) and cross-sectional radius \(r\) at its brim is given by \(V=\frac{1}{3} \pi r^{2} h .\).)
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