Textbook Question
Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.
f(x) = x²e⁻ˣ
Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 4.1.65
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = x³e⁻ˣ on [-1,5]
Verified step by step guidance1
First, understand that absolute maxima and minima refer to the highest and lowest points on the graph of a function within a given interval. To find these, we need to evaluate the function at critical points and endpoints of the interval.
To find critical points, we need to take the derivative of the function ƒ(x) = x³e⁻ˣ. Use the product rule for differentiation: if u(x) = x³ and v(x) = e⁻ˣ, then ƒ'(x) = u'(x)v(x) + u(x)v'(x).
Calculate the derivative: u'(x) = 3x² and v'(x) = -e⁻ˣ. Therefore, ƒ'(x) = 3x²e⁻ˣ - x³e⁻ˣ.
Set the derivative ƒ'(x) = 0 to find critical points. This simplifies to x²(3 - x)e⁻ˣ = 0. Solve for x to find the critical points within the interval [-1, 5].
Evaluate the function ƒ(x) at the critical points found and at the endpoints x = -1 and x = 5. Compare these values to determine the absolute maximum and minimum values on the interval.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Critical Points
Critical points are values of x in the domain of a function where the derivative is either zero or undefined. These points are essential for finding absolute maxima and minima, as they indicate where the function's slope changes, potentially leading to extreme values. To locate critical points, one must first compute the derivative of the function and solve for x.
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Critical Points
Endpoints of the Interval
When determining absolute extrema on a closed interval, it is crucial to evaluate the function at both the critical points and the endpoints of the interval. The absolute maximum or minimum could occur at any of these locations. In this case, the interval is [-1, 5], so the function must be evaluated at x = -1 and x = 5 in addition to any critical points found.
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First Derivative Test
The First Derivative Test is a method used to determine whether a critical point is a local maximum, local minimum, or neither. By analyzing the sign of the derivative before and after the critical point, one can infer the behavior of the function. If the derivative changes from positive to negative, the critical point is a local maximum; if it changes from negative to positive, it is a local minimum.
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07:09The First Derivative Test: Finding Local Extrema
Related Practice
Textbook Question
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = (2x)ˣ on [0.1,1]
Textbook Question
{Use of Tech} Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible.
x²/₃ + y²/₃ = 1 (Astroid or hypocycloid with four cusps)
Textbook Question
Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.
ƒ(x) = sec x on [-(π/4),π/4]
Textbook Question
Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. <IMAGE>
Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0 (sin x - x) / 7x³