Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.4.119

Chapter 4, Problem 4.4.119

119. Find the values of constants a, b, and c such that the graph of y = ax^3 + bx^2 + cx has a
local maximum at x = 3, local minimum at x =- 1, and inflection point at (1, 11).

Verified step by step guidance

To find the values of a, b, and c, start by using the conditions for local maxima and minima. The first derivative of the function y = ax^3 + bx^2 + cx is y' = 3ax^2 + 2bx + c. Set y'(3) = 0 for the local maximum at x = 3, and y'(-1) = 0 for the local minimum at x = -1.

Next, use the condition for the inflection point. The second derivative of the function is y'' = 6ax + 2b. Set y''(1) = 0 to find the inflection point at x = 1.

Substitute x = 3 into y' = 3ax^2 + 2bx + c to get the equation 27a + 6b + c = 0. This is the first equation.

Substitute x = -1 into y' = 3ax^2 + 2bx + c to get the equation 3a - 2b + c = 0. This is the second equation.

Substitute x = 1 into y'' = 6ax + 2b to get the equation 6a + 2b = 0. This is the third equation. Solve this system of equations to find the values of a, b, and c.

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Key Concepts

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

Critical Points and Derivatives

Critical points occur where the first derivative of a function is zero or undefined, indicating potential local maxima or minima. To find these points for the function y = ax^3 + bx^2 + cx, compute the first derivative and set it to zero. Solving this will help identify the x-values where the local maximum and minimum occur, which are given as x = 3 and x = -1.

Second Derivative Test

The second derivative test helps determine the nature of critical points. If the second derivative at a critical point is positive, the point is a local minimum; if negative, it's a local maximum. For the function y = ax^3 + bx^2 + cx, evaluate the second derivative at x = 3 and x = -1 to confirm the local maximum and minimum, respectively.

Inflection Points

An inflection point is where the function changes concavity, identified by setting the second derivative to zero and confirming a sign change. For y = ax^3 + bx^2 + cx, the inflection point is given at (1, 11). Ensure the second derivative equals zero at x = 1 and verify the function's value at this point matches the y-coordinate of the inflection point.

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Critical Points

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Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

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