Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.1.27

Chapter 4, Problem 4.1.27

Absolute Extrema on Finite Closed Intervals

In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.

h(x) = ³√x, −1 ≤ x ≤ 8

Verified step by step guidance

First, understand that to find the absolute extrema (maximum and minimum values) of a function on a closed interval, we need to evaluate the function at critical points and endpoints of the interval.

Identify the critical points of the function h(x) = ³√x. To do this, find the derivative of h(x) and set it equal to zero. The derivative of h(x) = ³√x is h'(x) = (1/3)x^(-2/3).

Since h'(x) = (1/3)x^(-2/3) is never zero, there are no critical points where the derivative is zero. However, check where the derivative is undefined. In this case, h'(x) is undefined at x = 0, which is within the interval [-1, 8].

Evaluate the function h(x) at the endpoints of the interval and at any critical points found. Calculate h(-1), h(0), and h(8).

Compare the values obtained from evaluating h(x) at x = -1, x = 0, and x = 8 to determine the absolute maximum and minimum values. These values will indicate the absolute extrema on the interval [-1, 8].

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Absolute Extrema

Absolute extrema refer to the highest and lowest values a function attains on a given interval. To find these, evaluate the function at critical points and endpoints of the interval. The absolute maximum is the largest value, and the absolute minimum is the smallest value among these evaluations.

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are potential locations for local extrema. For the function h(x) = ³√x, the derivative is undefined at x = 0, making it a critical point to consider when finding absolute extrema on the interval.

Graphing Functions

Graphing a function helps visualize its behavior over an interval, including where it reaches its extrema. For h(x) = ³√x, plot the function from x = -1 to x = 8, marking the critical points and endpoints. This visual representation aids in identifying the coordinates of the absolute maximum and minimum values.

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Related Practice

Textbook Question

Checking Antiderivative Formulas

Verify the formulas in Exercises 57–62 by differentiation.

∫(3x + 5)⁻² dx = −(3x + 5)⁻¹/3 + C

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Textbook Question

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62. Production level Suppose that c(x)=x^3-20x^2 + 20,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items.

Textbook Question

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫(sin2x − csc²x)dx

Textbook Question

Checking the Mean Value Theorem

Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.

f(x) = {sinx / x, −π ≤ x < 0

0, x = 0

Textbook Question

Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫2x(1 − x⁻³) dx

Textbook Question

Finding Extrema from Graphs

In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.

f(x) = |x|, −1 < x < 2