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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 4.1.5
Chapter 4, Problem 4.1.5

Finding Extrema from Graphs


In Exercises 1–6, determine from the graph whether the function has any absolute extreme values on [a, b]. Then explain how your answer is consistent with Theorem 1.


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1
Examine the graph of the function y = g(x) over the interval [a, b]. Identify any points where the function reaches a maximum or minimum value within this interval.
Notice that the function is continuous on the interval [a, b] except at point c, where there is a discontinuity. The function is not defined at c, as indicated by the open circle.
Identify the endpoints of the interval, a and b, and evaluate the function at these points. These values are potential candidates for absolute extrema.
Look for any local extrema within the interval [a, b] by observing the behavior of the graph. A local maximum or minimum could occur at a point where the graph changes direction.
Apply Theorem 1, which states that if a function is continuous on a closed interval [a, b], it must have an absolute maximum and minimum on that interval. Since the function is not continuous at c, consider only the endpoints and any local extrema that are defined within the interval.

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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 or lowest points on a function within a given interval. An absolute maximum is the highest point, while an absolute minimum is the lowest. These points are crucial in understanding the overall behavior of a function on a specified interval, such as [a, b].
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Closed Interval

A closed interval [a, b] includes all the points between a and b, as well as the endpoints themselves. This is important when determining extrema because the function must be evaluated at the endpoints to ensure all possible extreme values are considered, especially if the function is continuous.
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Theorem 1 (Extreme Value Theorem)

The Extreme Value Theorem states that if a function is continuous on a closed interval [a, b], it must have both a maximum and minimum value on that interval. This theorem is essential for identifying extrema, as it guarantees their existence under the right conditions, guiding the analysis of the graph.
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Related Practice
Textbook Question

Roots (Zeros)


Show that the functions in Exercises 19–26 have exactly one zero in the given interval.


g(t) = 1/(1 − t) + √(1 + t) − 3.1, (−1, 1)

Textbook Question

Root Finding

5. Use Newton's method to find the positive fourth root of 2 by solving the equation x^4 -2 = 0. Start with x_0 = 1 and find x_2.

Textbook Question

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

71. y' = x(x² - 12)

Textbook Question

Root Finding

2. Use Newton's method to estimate the one real solution of x^3 +3x + 1 = 0. Start with x_0 = 0 and then find x_2.

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.


∫(1 − cot²x) dx

Textbook Question

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

d³y/dx³ = 6; y″(0) = −8, y′(0) = 0, y(0) = 5