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) = eˣ(x - 2)²
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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 4.1.59
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) = (4x³/3) + 5x² - 6x on [0,5]
Verified step by step guidance1
First, find the derivative of the function ƒ(x) = \( \frac{4x^3}{3} + 5x^2 - 6x \). This will help us identify critical points where the slope of the tangent is zero or undefined.
Set the derivative equal to zero to find the critical points. Solve the equation \( \frac{d}{dx} \left( \frac{4x^3}{3} + 5x^2 - 6x \right) = 0 \).
Evaluate the function ƒ(x) at the critical points found in the previous step, as well as at the endpoints of the interval [0, 5]. This will help determine the potential locations of absolute extrema.
Compare the values of ƒ(x) at the critical points and the endpoints. The largest value will be the absolute maximum, and the smallest value will be the absolute minimum on the interval.
Conclude by stating the location and value of the absolute maximum and minimum based on the evaluations from the previous step.
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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 local maxima and minima, as they indicate where the function's slope changes. To locate absolute extrema on a closed interval, one must evaluate the function at these critical points as well as at the endpoints of the interval.
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Critical Points
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
Endpoints of the Interval
When finding absolute extrema on a closed interval, it is crucial to evaluate the function at the endpoints of the interval in addition to the critical points. The absolute maximum or minimum could occur at these endpoints, especially if the function is not continuous or has no critical points within the interval. Thus, both critical points and endpoints must be considered to ensure all potential extrema are identified.
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05:59Estimating the Area Under a Curve with Right Endpoints & Midpoint
Related Practice
Textbook Question
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = 3x³ + 3x² / 2 - 2x
Textbook Question
{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.
f(x) = x ln (x + 1) -1 ; x₀ = 1.7
Textbook Question
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = eˣ/(e²ᵉ + 1)
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Textbook Question
Rectangles beneath a parabola A rectangle is constructed with its base on the x-axis and two of its vertices on the parabola y = 48 - x². What are the dimensions of the rectangle with the maximum area? What is the area?
Textbook Question
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(t) = t/ t² + 1