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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 4.3.54a
Chapter 4, Problem 4.3.54a

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

Verified step by step guidance
1
First, find the derivative of the function f(x) = sin(x) - cos(x). The derivative, f'(x), is obtained by differentiating each term: f'(x) = cos(x) + sin(x).
Next, set the derivative equal to zero to find the critical points: cos(x) + sin(x) = 0. Solve this equation for x within the interval [0, 2π].
To solve cos(x) + sin(x) = 0, you can rewrite it as sin(x) = -cos(x). This can be expressed as tan(x) = -1, which gives the solutions x = 3π/4 and x = 7π/4 within the interval [0, 2π].
Evaluate the function f(x) at the critical points and at the endpoints of the interval. Calculate f(0), f(3π/4), f(7π/4), and f(2π).
Compare the values obtained in the previous step to determine the local extrema. The largest value is the local maximum, and the smallest value is the local minimum within the interval [0, 2π].

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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. To find these points, one typically examines the derivative of the function to identify critical points where the derivative is zero or undefined, and then uses the second derivative test or evaluates the function at these points to determine the nature of the extrema.
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Finding Extrema Graphically

Derivative

The derivative of a function represents the rate at which the function's value changes with respect to changes in its input. It is a fundamental tool in calculus for analyzing the behavior of functions, particularly in finding critical points where the function's slope is zero, indicating potential local maxima or minima. For the function f(x) = sin x − cos x, the derivative helps identify where the function's slope changes direction.
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Derivatives

Interval Analysis

Interval analysis involves examining the behavior of a function within a specified range of input values. In the context of finding extrema, it is crucial to consider the endpoints of the interval as well as any critical points within it. For the function f(x) = sin x − cos x on the interval 0 ≤ x ≤ 2π, this means evaluating the function at x = 0, x = 2π, and any critical points found within this range to determine where local extrema occur.
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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

Analyzing Functions from Derivatives


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


a. What are the critical points of f?


f′(x) = x(x − 1)

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 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

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