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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.1.60b
Chapter 4, Problem 4.1.60b

Theory and Examples


Cubic functions Consider the cubic function f(x) = ax³ + bx² + cx + d.


b. How many local extreme values can f have?

Verified step by step guidance
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To determine the number of local extreme values a cubic function can have, we first need to find its critical points. Critical points occur where the derivative of the function is zero or undefined.
Calculate the derivative of the cubic function f(x) = ax³ + bx² + cx + d. The derivative, f'(x), is given by: 3ax2+2bx+c.
Set the derivative f'(x) equal to zero to find the critical points: 3ax2+2bx+c=0. Solve this quadratic equation for x.
A quadratic equation can have at most two real roots, which means the cubic function can have at most two critical points. Each critical point is a candidate for a local extreme value.
To determine if these critical points are local maxima or minima, use the second derivative test. Calculate the second derivative, f''(x), and evaluate it at each critical point. If f''(x) is positive at a critical point, it is a local minimum; if negative, it is a local maximum.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cubic Functions

Cubic functions are polynomial functions of degree three, represented in the form f(x) = ax³ + bx² + cx + d, where a, b, c, and d are constants and a ≠ 0. The shape of the graph can vary significantly based on the coefficients, leading to different behaviors in terms of local extrema and inflection points.
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Properties of Functions

Local Extreme Values

Local extreme values refer to points in the function where it reaches a local maximum or minimum. For cubic functions, these points occur where the first derivative of the function equals zero, indicating potential changes in the direction of the graph. The number of local extrema can vary based on the function's specific coefficients.
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Average Value of a Function

First Derivative Test

The First Derivative Test is a method used to determine the local extrema of a function. By finding the derivative of the function and setting it to zero, we can identify critical points. Analyzing the sign of the derivative before and after these points helps classify them as local maxima, minima, or points of inflection.
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Related Practice
Textbook Question

Roots (Zeros)


a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.


iii. y = x³ − 3x² + 4 = (x + 1)(x − 2)²

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Textbook Question

Roots (Zeros)


a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.


iv. y = x³ − 33x² + 216x = x(x - 9)(x − 24)

Textbook Question

Roots (Zeros)


a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.


i. y = x² − 4

Textbook Question

Theory and Examples


In Exercises 51 and 52, give reasons for your answers.


Let f(x) = (x − 2)²ᐟ³.


b. Show that the only local extreme value of f occurs at x = 2.

Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

-(1/2)x⁻³ᐟ²

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Textbook Question

Analyzing Functions from Derivatives


Answer the following questions about the functions whose derivatives are given in Exercises 1–14:


b. On what open intervals is f increasing or decreasing?


f′(x) = x(x − 1)