Textbook Question
80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.
b. f(x) = x, g(x) = x², (a, b) arbitrary
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.3.131b
131. Let f(x) = x * e^(−x).
b. Find all inflection points for f.
Verified step by step guidance1
Recall that inflection points occur where the concavity of the function changes, which means where the second derivative changes sign. So, first, we need to find the second derivative of the function \(f(x) = x \cdot e^{-x}\).
Start by finding the first derivative \(f'(x)\). Use the product rule: if \(f(x) = u(x) \cdot v(x)\), then \(f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)\). Here, \(u(x) = x\) and \(v(x) = e^{-x}\).
Calculate \(u'(x) = 1\) and \(v'(x) = -e^{-x}\) (using the chain rule). Substitute these into the product rule formula to get \(f'(x) = 1 \cdot e^{-x} + x \cdot (-e^{-x})\).
Simplify \(f'(x)\) to \(f'(x) = e^{-x} - x e^{-x} = e^{-x}(1 - x)\). Next, find the second derivative \(f''(x)\) by differentiating \(f'(x)\) again, applying the product rule to \(e^{-x}(1 - x)\).
Set \(f''(x) = 0\) and solve for \(x\) to find potential inflection points. Then, check the sign of \(f''(x)\) on intervals around these points to confirm where the concavity changes, identifying the inflection points.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Second Derivative and Concavity
The second derivative of a function measures the curvature or concavity of its graph. If the second derivative is positive on an interval, the graph is concave up; if negative, concave down. Inflection points occur where the concavity changes, typically where the second derivative equals zero or is undefined.
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The Second Derivative Test: Finding Local Extrema
Finding Inflection Points
To find inflection points, first compute the second derivative of the function, then solve for points where it equals zero or does not exist. Finally, verify that the concavity changes sign around these points to confirm they are true inflection points.
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04:50Critical Points
Product Rule for Differentiation
Since f(x) = x * e^(−x) is a product of two functions, the product rule is used to find its derivatives. The product rule states that the derivative of u(x)v(x) is u'(x)v(x) + u(x)v'(x), which is essential for correctly computing the first and second derivatives.
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05:18The Product Rule
Related Practice
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
8. b. arccot(√3)
Textbook Question
132. Let f(x) = e^x / (1 + e^(2x)).
b. Find all inflection points for f.
Textbook Question
In Exercises 1–4, solve for t.
2. b. e^(kt) = 10
Textbook Question
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.
72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2
Textbook Question
71. Locate and identify the absolute extreme values of cos(ln x) on [1/2, 2]