Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problema 4.1.60b

Capitolo 4, Problema 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?

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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:

3 a x^{2} + 2 b x + c

Set the derivative f'(x) equal to zero to find the critical points:

3 a x^{2} + 2 b x + 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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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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Percorso guidato

06:21

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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Percorso guidato

06:37

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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07:09

The First Derivative Test: Finding Local Extrema

Pratica correlata

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Checking Antiderivative Formulas

Right, or wrong? Say which for each formula and give a brief reason for each answer.

∫xsinx dx = -x cos x + C

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[Technology Exercise] 17. Designing a suitcase A 24-in.-by-36-in. sheet of cardboard is folded in half to form a 24-in.-by-18-in. rectangle as shown in the accompanying figure. Then four congruent squares of side length x are cut from the corners of the folded rectangle. The sheet is unfolded, and the six tabs are folded up to form a box with sides and a lid.

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b. Find the domain of V for the problem situation and graph V over this domain.

170

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38. What values of a and b make f(x) = x^3 + ax^2 + bx have

b. a local minimum at x = 4 and a point of inflection at x = 1?

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Analyzing Functions from Derivatives

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