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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.1.37
Chapter 4, Problem 4.1.37

Absolute Extrema on Finite Closed Intervals


In Exercises 37–40, find the function’s absolute maximum and minimum values and say where they occur.


f(x) = x⁴ᐟ³, −1 ≤ x ≤ 8

Verified step by step guidance
1
Identify the function f(x) = x^(4/3) and the closed interval [-1, 8]. We need to find the absolute maximum and minimum values of this function on the given interval.
Calculate the derivative of the function to find the critical points. The derivative f'(x) is found using the power rule: f'(x) = (4/3)x^(1/3).
Set the derivative equal to zero to find critical points: (4/3)x^(1/3) = 0. Solve for x to find the critical points within the interval.
Evaluate the function f(x) at the critical points found in the previous step, as well as at the endpoints of the interval, x = -1 and x = 8.
Compare the values of f(x) at the critical points and the endpoints to determine the absolute maximum and minimum values 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 can achieve on a given interval. To find these values, 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 within the specified range.
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Finding Extrema Graphically

Critical Points

Critical points are values of x where the derivative of the function is zero or undefined. These points are potential candidates for local maxima or minima. To find critical points, take the derivative of the function, set it equal to zero, and solve for x. Also, check where the derivative does not exist within the interval.
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Evaluating Endpoints

Evaluating endpoints involves calculating the function's value at the boundaries of the interval. For the function f(x) = x⁴ᐟ³ on the interval [-1, 8], compute f(-1) and f(8). These values, along with those at critical points, help determine the absolute extrema by comparing them to find the maximum and minimum values.
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Related Practice
Textbook Question

Applications


A marathoner ran the 26.2-mi New York City Marathon in 2.2 hours. Show that at least twice the marathoner was running at exactly 11 mph, assuming the initial and final speeds are zero.

Textbook Question

The 8-ft wall shown here stands 27 ft from the building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall.

Textbook Question

Checking Antiderivative Formulas


Verify the formulas in Exercises 57–62 by differentiation.


∫csc²((x − 1)/3)dx = −3cot((x − 1)/3) + C

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