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Chapter 4: Problem 2

Suppose \(f\) is a continuous function defined on a closed interval \([a, b] .\) (a) What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for \(f\)? (b) What steps would you take to find those maximum and minimum values?

Short Answer

Expert verified
(a) Extreme Value Theorem. (b) Find the derivative, evaluate at critical points and endpoints, then compare.

Step by step solution

01

Identifying the Theorem

The theorem that guarantees the existence of an absolute maximum and minimum value for a continuous function on a closed interval is the Extreme Value Theorem. The Extreme Value Theorem states that if a function is continuous on a closed interval \([a, b]\), then the function must attain a maximum and a minimum value within that interval.
02

Determine the Critical Points

To find the maximum and minimum values of the function \(f\) on the interval \([a, b]\), begin by finding the derivative \(f'(x)\). Set the derivative equal to zero and solve for \(x\) to find the critical points, i.e., solve the equation \ f'(x) = 0 \.
03

Evaluate at Endpoints

Evaluate the function \(f(x)\) at the endpoints of the interval \([a, b]\)鈥攕pecifically at \(x = a\) and \(x = b\). These values are necessary as extrema can occur at the endpoints.
04

Evaluate at Critical Points

Evaluate the function \(f(x)\) at the critical points found in Step 1. These evaluations will give the potential candidates for local maxima or minima within the interval.
05

Compare Values to Find Extremes

Compare the values of \(f(x)\) obtained from evaluating at the critical points and endpoints. The largest value will correspond to the absolute maximum, and the smallest value will correspond to the absolute minimum on \([a, b]\).

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Key Concepts

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

Absolute Maximum
The concept of an "absolute maximum" is a fundamental topic in calculus. It refers to the highest value a function attains on a given interval. For a continuous function defined on a closed interval \([a, b]\), the absolute maximum is the largest output value it produces in that interval.

To identify the absolute maximum of a function \( f(x) \), we need to evaluate the function at various critical points, including:
  • Endpoints of the interval (\(a\) and \(b\))
  • Critical points inside the interval where the derivative \(f'(x)\) is zero or undefined
Evaluate the function at each of these points and compare the values. The point with the highest value is the absolute maximum. It ensures that students can effectively find the peak output value on their respective interval.
Absolute Minimum
Similar to the absolute maximum, the "absolute minimum" of a function is the lowest value that the function achieves on a given interval. It represents the smallest output for the function on \( [a, b] \).

To find the absolute minimum, compare the function's values at comprised:
  • The endpoints of the interval \(a\) and \(b\)
  • All critical points where \(f'(x) = 0\) or is undefined
Calculate the function at these points, and the smallest value obtained is the absolute minimum. Understanding this concept is crucial for analyzing the full range of a function's output.
Critical Points
Critical points are essential in determining where a function's high and low points lie. A critical point occurs if the derivative of the function is either zero or undefined. These points mark potential locations for local maxima and minima.

To find critical points:
  • Start by finding the derivative of the function, \( f'(x) \).
  • Set the derivative equal to zero, \(f'(x) = 0\), and solve for \(x\).
  • Check where the derivative might be undefined.
Critical points are crucial because they might indicate where the function changes direction, potentially leading to maximum or minimum values in the interval. These points, along with endpoints, are considered when finding absolute extrema.
Continuous Function
A continuous function is a fundamental concept in calculus, ensuring no breaks, jumps, or holes in its graph. For the Extreme Value Theorem to apply, a function must be continuous on a closed interval \( [a, b] \). This continuity guarantees the function attains both absolute maximum and minimum values within the interval.

Continuous functions on closed intervals allow us to:
  • Apply the Extreme Value Theorem effectively
  • Ensure that endpoints and critical points can be safely evaluated
  • Confidently compare calculated values to determine extrema
The smoothness and consistency of a continuous function are what make it a reliable candidate for analysis, as it provides certainty that essential values do, indeed, exist on the specified interval.

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Most popular questions from this chapter

A woman at a point \(A\) on the shore of a circular lake with radius 2 mi wants to arrive at the point \(C\) diametrically oppo- site \(A\) on the other side of the lake in the shortest possible time (see the figure). She can walk at the rate of 4\(\mathrm { mi } / \mathrm { h }\) and row a boat at 2\(\mathrm { mi } / \mathrm { h }\) . How should she proceed?

Use a computer algebra system to graph \(f\) and to find \(f^{\prime}\) and \(f^{\prime \prime} .\) Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of \(f\). \(f(x)=\left(x^{2}-1\right) e^{\arctan x}\)

Use a computer algebra system to graph \(f\) and to find \(f^{\prime}\) and \(f^{\prime \prime} .\) Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of \(f\). \(f(x)=\sqrt{x+5 \sin x}, \quad x \leqslant 20\)

A rain gutter is to be constructed from a metal sheet of width 30\(\mathrm { cm }\) by bending up one-third of the sheet on each side through an angle \(\theta\) . How should \(\theta\) be chosen so that the gutter will carry the maximum amount of water?

Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than over land because air generally rises over land and falls over water during the day. A bird with these tendencies is released from an island that is 5\(\mathrm { km }\) from the nearest point \(B\) on a straight shoreline, flies to a point \(C\) on the shoreline, and then flies along the shoreline to its nesting area \(D .\) Assume that the bird instinctively chooses a path that will minimize its energy expenditure. Points \(B\) and \(D\) are 13\(\mathrm { km }\) apart. (a) In general, if it takes 1.4 times as much energy to fly over water as it does over land, to what point \(C\) should the bird fly in order to minimize the total energy expended in returning to its nesting area? (b) Let \(W\) and \(L\) denote the energy (in joules) per kilometer flown over water and land, respectively. What would a large value of the ratio \(W / L\) mean in terms of the bird's flight? What would a small value mean? Determine the ratio \(W / L\) corresponding to the minimum expenditure of energy. (c) What should the value of \(W / L\) be in order for the bird to fly directly to its nesting area \(D ?\) What should the value of \(W / L\) be for the bird to fly to \(B\) and then along the shore to \(D ?\) (d) If the omithologists observe that birds of a certain species reach the shore at a point 4\(\mathrm { km }\) from \(B\) , how many times more energy does it take a bird to fly over water than over land?
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