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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 4.5.38b
Chapter 4, Problem 4.5.38b

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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1
To find the values of a and b, we need to use the conditions given: a local minimum at x = 4 and a point of inflection at x = 1.
First, find the first derivative of f(x): f'(x) = 3x^2 + 2ax + b. For a local minimum at x = 4, f'(4) must be 0.
Substitute x = 4 into f'(x) to get the equation: 3(4)^2 + 2a(4) + b = 0. Simplify this to find one equation in terms of a and b.
Next, find the second derivative of f(x): f''(x) = 6x + 2a. For a point of inflection at x = 1, f''(1) must be 0.
Substitute x = 1 into f''(x) to get the equation: 6(1) + 2a = 0. Solve this equation to find the value of a, and then use it in the first equation to find b.

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Key Concepts

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

Local Minimum

A local minimum of a function occurs at a point where the function value is lower than at nearby points. To find a local minimum, the first derivative of the function is set to zero, indicating a critical point, and the second derivative is checked to be positive, confirming a local minimum. In this problem, the function f(x) must have a local minimum at x = 4.
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The Second Derivative Test: Finding Local Extrema

Point of Inflection

A point of inflection is where the function's concavity changes, which can be identified by the second derivative. At a point of inflection, the second derivative is zero, but the sign of the second derivative changes around this point. For the function f(x), there must be a point of inflection at x = 1, meaning the second derivative changes sign at this point.
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Critical Points

Derivatives

Derivatives are fundamental in calculus for analyzing the behavior of functions. The first derivative provides information about the slope and critical points, while the second derivative gives insights into concavity and points of inflection. Solving the problem requires calculating both the first and second derivatives of f(x) = x^3 + ax^2 + bx to find values of a and b that satisfy the given conditions.
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Derivatives
Related Practice
Textbook Question

Checking Antiderivative Formulas


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


∫3(2x + 1)² dx = (2x + 1)³ + C

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Textbook Question

Theory and Examples


Sketch the graph of a differentiable function y = f(x) that has a local maximum at (1, 1) and a local minimum at (3, 3).

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.

4 sec 3x tan 3x

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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)

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.

x⁷

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