Domanda del libro di testo
117. Suppose that the second derivative of the function y = f(x) isy" =(x+1)(x-2).
For what x-values does the graph of f have an inflection point?
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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 4.2.6
Checking the Mean Value Theorem
Find the value or values of c that satisfy the equation (f(b) − f(a)) / (b − a) = f′(c) in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–6.
g(x) = {x³, −2 ≤ x ≤ 0
x², 0 < x ≤ 2
Guida verificata passo dopo passo1
First, understand the Mean Value Theorem (MVT). It states that for a function g(x) that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), there exists at least one c in (a, b) such that (g(b) - g(a)) / (b - a) = g'(c).
Identify the intervals and the function pieces: g(x) = x³ for -2 ≤ x ≤ 0 and g(x) = x² for 0 < x ≤ 2. The function is continuous on [-2, 2] but we need to check differentiability at x = 0.
Check differentiability at x = 0 by finding the derivatives of each piece: For x³, the derivative is g'(x) = 3x², and for x², the derivative is g'(x) = 2x. Evaluate the left-hand and right-hand derivatives at x = 0 to ensure they are equal.
Calculate g(-2) and g(2) to apply the MVT: g(-2) = (-2)³ = -8 and g(2) = (2)² = 4. Use these to find the average rate of change: (g(2) - g(-2)) / (2 - (-2)) = (4 - (-8)) / 4.
Set the average rate of change equal to the derivative g'(c) and solve for c in each interval: For -2 < c < 0, use g'(c) = 3c², and for 0 < c < 2, use g'(c) = 2c. Solve these equations to find the value(s) of c that satisfy the MVT.
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Mean Value Theorem
The Mean Value Theorem (MVT) states that for a continuous function on a closed interval [a, b] that is differentiable on the open interval (a, b), there exists at least one point c in (a, b) such that the derivative at c equals the average rate of change over [a, b]. This is expressed as f'(c) = (f(b) - f(a)) / (b - a).
Video consigliato:
Percorso guidato
06:11
Fundamental Theorem of Calculus Part 1
Piecewise Functions
A piecewise function is defined by different expressions over different intervals. In this problem, g(x) is defined as x³ for -2 ≤ x ≤ 0 and x² for 0 < x ≤ 2. Understanding how to evaluate and differentiate each piece within its respective interval is crucial for applying the Mean Value Theorem.
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Percorso guidato
05:36Piecewise Functions
Differentiability and Continuity
For the Mean Value Theorem to apply, the function must be continuous on the closed interval and differentiable on the open interval. This requires checking that g(x) is continuous at x = 0, where the pieces meet, and ensuring differentiability across the entire interval, except possibly at the endpoints.
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05:34Intro to Continuity
Pratica correlata
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Initial Value Problems
Solve the initial value problems in Exercises 71–90.
dv/dt = (1/2)sec t tan t, v(0) = 1
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Identifying Extrema
In Exercises 19–40:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local extreme values, if any, saying where they occur.
f(x) = x³ / (3x² + 1)
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Domanda del libro di testo
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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Domanda del libro di testo
"Roots (Zeros) Show that the functions in Exercises 19–26 have exactly one zero
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Domanda del libro di testo
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
y = x² − 32√x
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