Domanda del libro di testo
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) = sec²x − 2tan x, −π/2 < x < π/2
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 4, Problema 4.3.55a
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) = √3cos x + sin x, 0 ≤ x ≤ 2π
Guida verificata passo dopo passo1
To find the local extrema of the function \( f(x) = \sqrt{3} \cos x + \sin x \) on the interval \( 0 \leq x \leq 2\pi \), start by finding the derivative of the function. The derivative \( f'(x) \) is given by \( f'(x) = -\sqrt{3} \sin x + \cos x \).
Set the derivative equal to zero to find the critical points: \( -\sqrt{3} \sin x + \cos x = 0 \). Rearrange this equation to \( \cos x = \sqrt{3} \sin x \).
Divide both sides by \( \cos x \) (assuming \( \cos x \neq 0 \)) to get \( 1 = \sqrt{3} \tan x \), which simplifies to \( \tan x = \frac{1}{\sqrt{3}} \).
Solve \( \tan x = \frac{1}{\sqrt{3}} \) for \( x \) within the interval \( 0 \leq x \leq 2\pi \). The solutions are \( x = \frac{\pi}{6} \) and \( x = \frac{7\pi}{6} \).
Evaluate the function \( f(x) \) at the critical points \( x = \frac{\pi}{6} \), \( x = \frac{7\pi}{6} \), and the endpoints \( x = 0 \) and \( x = 2\pi \) to determine the local extrema. Compare these values to identify the local maxima and minima.
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Local Extrema
Local extrema refer to the points in a function where it reaches a local maximum or minimum value. These are points where the function changes direction, and they can be found by analyzing the first derivative. A local maximum occurs when the derivative changes from positive to negative, while a local minimum occurs when it changes from negative to positive.
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05:58
Finding Extrema Graphically
First Derivative Test
The first derivative test is a method used to identify local extrema of a function. By taking the derivative of the function and finding its critical points (where the derivative is zero or undefined), we can determine where the function's slope changes. Analyzing the sign of the derivative before and after these points helps identify whether they are local maxima or minima.
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07:09The First Derivative Test: Finding Local Extrema
Trigonometric Functions
Trigonometric functions, such as sine and cosine, are periodic functions that describe oscillatory behavior. In the context of finding extrema, understanding their periodicity and behavior over specific intervals, like 0 to 2π, is crucial. These functions have known maximum and minimum values, which can help in determining the extrema of more complex functions involving them.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Pratica correlata
Domanda del libro di testo
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = (x − 2)²ᐟ³.
a. Does f′(2) exist?
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Domanda del libro di testo
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 2x, 0 ≤ x ≤ π
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Domanda del libro di testo
Identifying Extrema
In Exercises 41–52:
a. Identify the function’s local extreme values in the given domain, and say where they occur.
k(x) = x³ + 3x² + 3x + 1, −∞ < x ≤ 0
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Domanda del libro di testo
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
206
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Domanda del libro di testo
20.The U.S. Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. a.What dimensions will give a box with a square end the largest possible volume?
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