교과서 질문
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 ― 2𝓍 + 4
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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
4장, 문제 4.1.37
Absolute Extrema on Finite Closed Intervals
In Exercises 37–40, find the function’s absolute maximum and minimum values and say where they occur.
f(x) = x⁴ᐟ³, −1 ≤ x ≤ 8
검증된 단계별 안내1
Identify the function f(x) = x^(4/3) and the closed interval [-1, 8]. We need to find the absolute maximum and minimum values of this function on the given interval.
Calculate the derivative of the function to find the critical points. The derivative f'(x) is found using the power rule: f'(x) = (4/3)x^(1/3).
Set the derivative equal to zero to find critical points: (4/3)x^(1/3) = 0. Solve for x to find the critical points within the interval.
Evaluate the function f(x) at the critical points found in the previous step, as well as at the endpoints of the interval, x = -1 and x = 8.
Compare the values of f(x) at the critical points and the endpoints to determine the absolute maximum and minimum values on the interval [-1, 8].
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주요 개념
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Absolute Extrema
Absolute extrema refer to the highest and lowest values a function can achieve on a given interval. To find these values, evaluate the function at critical points and endpoints of the interval. The absolute maximum is the largest value, and the absolute minimum is the smallest value within the specified range.
추천 영상:
05:58
Finding Extrema Graphically
Critical Points
Critical points are values of x where the derivative of the function is zero or undefined. These points are potential candidates for local maxima or minima. To find critical points, take the derivative of the function, set it equal to zero, and solve for x. Also, check where the derivative does not exist within the interval.
추천 영상:
04:50Critical Points
Evaluating Endpoints
Evaluating endpoints involves calculating the function's value at the boundaries of the interval. For the function f(x) = x⁴ᐟ³ on the interval [-1, 8], compute f(-1) and f(8). These values, along with those at critical points, help determine the absolute extrema by comparing them to find the maximum and minimum values.
추천 영상:
5:14Evaluate Logarithms
관련 실천
교과서 질문
In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
53. y = x * √(8 - x²)
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교과서 질문
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
y = x² + 2/x
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교과서 질문
"Roots (Zeros) Show that the functions in Exercises 19–26 have exactly one zero
206
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교과서 질문
Solve the initial value problems in Exercises 71–90.
y⁽⁴⁾ = −sin t + cos t;
y′′′(0) =7, y′′(0) = y′(0) = −1, y(0) = 0
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교과서 질문
Business and Economics
60. Production level Prove that the production level (if any) at which average cost is smallest is a level at which the average cost equals marginal cost.
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