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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.1.51b
Capitolo 4, Problema 4.1.51b

Theory and Examples


In Exercises 51 and 52, give reasons for your answers.


Let f(x) = (x − 2)²ᐟ³.


b. Show that the only local extreme value of f occurs at x = 2.

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1
To find local extreme values, we need to determine where the derivative of the function f(x) = (x - 2)^(2/3) is zero or undefined. Start by finding the derivative f'(x).
Use the chain rule to differentiate f(x). Let u = x - 2, then f(x) = u^(2/3). The derivative f'(x) = (2/3) * u^(-1/3) * du/dx, where du/dx = 1.
Simplify the derivative: f'(x) = (2/3) * (x - 2)^(-1/3). This derivative is undefined at x = 2 because the expression (x - 2)^(-1/3) involves division by zero.
Since f'(x) is undefined at x = 2, check the behavior of f(x) around x = 2 to determine if it is a local extreme value. Consider the sign of f'(x) as x approaches 2 from the left and right.
Observe that as x approaches 2 from the left, f'(x) is negative, and as x approaches 2 from the right, f'(x) is positive. This change in sign indicates that x = 2 is a local minimum.

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Local Extreme Values

Local extreme values refer to points in the domain of a function where the function takes on a maximum or minimum value relative to nearby points. To identify these points, we typically use the first derivative test, which involves finding critical points where the derivative is zero or undefined, and then analyzing the behavior of the function around these points.
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Average Value of a Function

First Derivative Test

The first derivative test is a method used to determine whether a critical point is a local maximum, local minimum, or neither. By evaluating the sign of the derivative before and after the critical point, we can conclude if the function is increasing or decreasing, thus identifying the nature of the extreme value at that point.
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The First Derivative Test: Finding Local Extrema

Critical Points

Critical points of a function occur where its derivative is either zero or undefined. These points are essential for finding local extreme values, as they represent potential locations where the function's behavior changes. In the context of the given function f(x) = (x − 2)²/3, identifying critical points will help in determining where the local extreme value occurs.
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Critical Points
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Identifying Extrema


In Exercises 19–40:


b. Identify the function’s local extreme values, if any, saying where they occur.


f(θ) = 3θ² − 4θ³

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Checking Antiderivative Formulas


Right, or wrong? Say which for each formula and give a brief reason for each answer.


∫3(2x + 1)² dx = (2x + 1)³ + C

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Domanda del libro di testo

Identifying Extrema


In Exercises 19–40:


b. Identify the function’s local extreme values, if any, saying where they occur.


f(r) = 3r³ + 16r

132
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Domanda del libro di testo

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

4 sec 3x tan 3x

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Domanda del libro di testo

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

x⁻³/2 + x²

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Domanda del libro di testo

Analyzing Functions from Derivatives


Answer the following questions about the functions whose derivatives are given in Exercises 1–14:



b. On what open intervals is f increasing or decreasing?


f′(x) = (x − 1)²(x + 2)

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