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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 50
Chapter 4, Problem 50

First Derivative Test


a. Locate the critical points of f.
b. Use the First Derivative Test to locate the local maximum and minimum values.
c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).


f(x) = 2x⁵ - 5x⁴ - 10x³ + 4 on [-2,4]

Verified step by step guidance
1
To find the critical points of the function \( f(x) = 2x^5 - 5x^4 - 10x^3 + 4 \), first compute the derivative \( f'(x) \). Use the power rule to differentiate each term: \( f'(x) = 10x^4 - 20x^3 - 30x^2 \).
Set the derivative \( f'(x) \) equal to zero to find the critical points: \( 10x^4 - 20x^3 - 30x^2 = 0 \). Factor the equation to solve for \( x \).
Factor out the greatest common factor: \( 10x^2(x^2 - 2x - 3) = 0 \). Solve \( 10x^2 = 0 \) to find \( x = 0 \). Then solve \( x^2 - 2x - 3 = 0 \) using the quadratic formula or factoring to find additional critical points.
Use the First Derivative Test to determine the nature of each critical point. Evaluate \( f'(x) \) on intervals around each critical point to determine if the function is increasing or decreasing, which will indicate local maxima or minima.
To find the absolute maximum and minimum values on the interval \([-2, 4]\), evaluate \( f(x) \) at the critical points and the endpoints \( x = -2 \) and \( x = 4 \). Compare these values to determine the absolute extrema.

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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 of a function occur where its derivative is zero or undefined. These points are essential for identifying local maxima and minima, as they represent potential locations where the function's behavior changes. To find critical points, one must first compute the derivative of the function and solve for the values of x that satisfy the condition f'(x) = 0.
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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 conclude that if the derivative changes from positive to negative, the point is a local maximum; if it changes from negative to positive, it is a local minimum.
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The First Derivative Test: Finding Local Extrema

Absolute Maximum and Minimum

The absolute maximum and minimum values of a function on a closed interval are the highest and lowest values the function attains within that interval. To find these values, one must evaluate the function at its critical points and at the endpoints of the interval. The largest and smallest of these values will be the absolute maximum and minimum, respectively.
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Finding Extrema Graphically Example 4
Related Practice
Textbook Question

Travel costs A simple model for travel costs involves the cost of gasoline and the cost of a driver. Specifically, assume gasoline costs \(p/gallon and the vehicle gets g miles per gallon. Also assume the driver earns \)w/hour.


e. Should the optimal speed be increased or decreased (compared with part (d)) if L is increased from 400 mi to 500 mi? Explain.

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Textbook Question

Mean Value Theorem The population of a culture of cells grows according to the function P(t) = 100t / t+1, where t ≥ 0 is measured in weeks.


b. At what point of the interval [0, 8] is the instantaneous rate of change equal to the average rate of change?

Textbook Question

Find the height h, radius r, and volume of a right circular cylinder with maximum volume that is inscribed in a sphere of radius R.

Textbook Question

First Derivative Test


a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).


f(x) = x²/(x² - 1) on [-4,4]

1
views
Textbook Question

Mean Value Theorem The population of a culture of cells grows according to the function P(t) = 100t / t+1, where t ≥ 0 is measured in weeks.




a. What is the average rate of change in the population over the interval [0, 8]?

Textbook Question

Travel costs A simple model for travel costs involves the cost of gasoline and the cost of a driver. Specifically, assume gasoline costs \(p/gallon and the vehicle gets g miles per gallon. Also assume the driver earns \)w/hour.

b. At what speed does the gas mileage function have its maximum?