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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 4.3.42a
Chapter 4, Problem 4.3.42a

Identifying Extrema


In Exercises 41–52:


a. Identify the function’s local extreme values in the given domain, and say where they occur.


f(x) = (x + 1)², −∞ < x ≤ 0

Verified step by step guidance
1
First, understand that local extrema refer to the local minimum or maximum values of a function within a given domain. For the function f(x) = (x + 1)^2, we need to find these values for -∞ < x ≤ 0.
To find the local extrema, we first need to find the derivative of the function f(x). The derivative, f'(x), will help us identify critical points where the slope of the tangent is zero or undefined.
Calculate the derivative of f(x) = (x + 1)^2. Using the power rule, the derivative is f'(x) = 2(x + 1).
Set the derivative equal to zero to find the critical points: 2(x + 1) = 0. Solve for x to find the critical point.
Evaluate the function f(x) at the critical point and at the endpoints of the domain (if applicable) to determine the local extrema. Since the domain is -∞ < x ≤ 0, consider the endpoint x = 0 as well.

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Key Concepts

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

Local Extrema

Local extrema refer to the points in a function where it reaches a local maximum or minimum within a specific interval. These are points where the function changes direction, and can be identified by analyzing the derivative of the function. In the context of the given domain, local extrema are found by evaluating the function's behavior at critical points and endpoints.
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Finding Extrema Graphically

Derivative and Critical Points

The derivative of a function provides information about its rate of change and helps identify critical points where the derivative is zero or undefined. These points are potential locations for local extrema. For the function f(x) = (x + 1)², the derivative f'(x) = 2(x + 1) can be used to find critical points by setting it equal to zero and solving for x.
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Critical Points

Domain Considerations

The domain of a function specifies the set of input values for which the function is defined. In this problem, the domain is −∞ < x ≤ 0, which affects where extrema can occur. It's crucial to consider endpoints and any restrictions in the domain when identifying extrema, as they can influence the function's behavior and the location of extrema.
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Finding the Domain and Range of a Graph
Related Practice
Textbook Question

Initial Value Problems

Solve the initial value problems in Exercises 89–92.

d^3 r/dt^3 = - cos t; r''(0) = r'(0) = 0 , r(0) = -1

Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ 1/( r + 5)²dr

Textbook Question

Finding Antiderivatives

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

sec²x

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Textbook Question

Identifying Extrema


In Exercises 53–60:


a. Find the local extrema of each function on the given interval, and say where they occur.


f(x) = sin x − cos x,0 ≤ x ≤ 2π

Textbook Question

Identifying Extrema


In Exercises 19–40:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


f(r) = 3r³ + 16r

Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ sec θ/3 tan θ/3 dθ