Domanda del libro di testo
Theory and Examples
[Technology Exercise] Graph the functions in Exercises 63–66. Then find the extreme values of the function on the interval and say where they occur.
f(x) = |x − 2| + |x + 3|, −5 ≤ x ≤ 5
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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
Finding Extreme Values
In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.
y = 𝓍³ + 𝓍² ― 8𝓍 + 5
Guida verificata passo dopo passo1
First, find the derivative of the function \( y = x^3 + x^2 - 8x + 5 \) to determine the critical points. The derivative is \( y' = 3x^2 + 2x - 8 \).
Set the derivative equal to zero to find the critical points: \( 3x^2 + 2x - 8 = 0 \). Solve this quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 3 \), \( b = 2 \), and \( c = -8 \).
Calculate the discriminant \( b^2 - 4ac \) to determine the nature of the roots. If the discriminant is positive, there are two distinct real roots, which are the critical points.
Evaluate the second derivative \( y'' = 6x + 2 \) at each critical point to determine the concavity and identify whether each critical point is a local maximum, local minimum, or a point of inflection.
Finally, analyze the behavior of the function as \( x \to \pm \infty \) to determine if there are any absolute extreme values over the natural domain. Consider the end behavior of the cubic function to conclude about absolute extrema.
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Critical Points
Critical points of a function occur where its derivative is zero or undefined. These points are potential locations for local extrema. To find them, compute the derivative of the function and solve for values of x where the derivative equals zero or does not exist. In this problem, finding the derivative of y = x³ + x² - 8x + 5 is essential to identify critical points.
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04:50
Critical Points
First Derivative Test
The First Derivative Test helps determine whether a critical point is a local maximum, minimum, or neither. By analyzing the sign changes of the derivative around the critical points, one can infer the nature of the extrema. If the derivative changes from positive to negative, the point is a local maximum; if it changes from negative to positive, it's a local minimum.
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07:09The First Derivative Test: Finding Local Extrema
Second Derivative Test
The Second Derivative Test provides another method to classify critical points. By evaluating the second derivative at a critical point, one can determine concavity: if the second derivative is positive, the function is concave up, indicating a local minimum; if negative, concave down, indicating a local maximum. This test complements the First Derivative Test for confirming extrema.
Video consigliato:
06:02The Second Derivative Test: Finding Local Extrema
Pratica correlata
Domanda del libro di testo
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)
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Domanda del libro di testo
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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Domanda del libro di testo
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.
74. y' = (x² - 2x)(x - 5)²
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Domanda del libro di testo
119. Find the values of constants a, b, and c such that the graph of y = ax^3 + bx^2 + cx has a
local maximum at x = 3, local minimum at x =- 1, and inflection point at (1, 11).
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Domanda del libro di testo
Theory and Examples
[Technology Exercise] Graph the functions in Exercises 63–66. Then find the extreme values of the function on the interval and say where they occur.
h(x) = |x + 2| − |x − 3|, −∞ < x < ∞
199
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