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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 4.4.113
Chapter 4, Problem 4.4.113

113. If b, c, and d are constants, for what value of b will the curve y = x^3 + bx^2 + cx + d have a
point of inflection at x = 1? Give reasons for your answer.

Verified step by step guidance
1
To find the point of inflection, we need to determine where the second derivative of the function changes sign. Start by finding the first derivative of the function y = x^3 + bx^2 + cx + d. The first derivative is y' = 3x^2 + 2bx + c.
Next, find the second derivative of the function. Differentiate the first derivative to get y'' = 6x + 2b.
A point of inflection occurs where the second derivative is zero or undefined, and changes sign. Set the second derivative equal to zero: 6x + 2b = 0.
Solve the equation 6x + 2b = 0 for x = 1, since the point of inflection is given to be at x = 1. Substitute x = 1 into the equation: 6(1) + 2b = 0.
Solve the equation 6 + 2b = 0 for b to find the value of b that ensures the curve has a point of inflection at x = 1.

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Key Concepts

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

Point of Inflection

A point of inflection on a curve is where the concavity changes from concave up to concave down, or vice versa. This occurs where the second derivative of the function changes sign. To find such points, we set the second derivative equal to zero and solve for the variable.
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Critical Points

Second Derivative

The second derivative of a function, denoted as f''(x), provides information about the concavity of the function. It is the derivative of the first derivative, f'(x). If f''(x) > 0, the function is concave up, and if f''(x) < 0, it is concave down. A change in sign of f''(x) indicates a potential point of inflection.
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The Second Derivative Test: Finding Local Extrema

Polynomial Derivatives

To find the derivatives of a polynomial function, apply the power rule: for a term ax^n, the derivative is n*ax^(n-1). For the function y = x^3 + bx^2 + cx + d, the first derivative is 3x^2 + 2bx + c, and the second derivative is 6x + 2b. These derivatives are used to analyze the behavior of the curve, such as finding points of inflection.
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Introduction to Polynomial Functions
Related Practice
Textbook Question

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

______

y = √𝓍² ― 1

Textbook Question

Initial Value Problems


Solve the initial value problems in Exercises 71–90.


d²r/dt² = 2/t³; dr/dt|ₜ ₌ ₁ =1, r(1) = 1

Textbook Question

Identifying Extrema


In Exercises 15–18:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


b. Identify the function’s local and absolute extreme values, if any, saying where they occur.


Textbook Question

Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫(−2cost) dt

Textbook Question

Checking the Mean Value Theorem


Find the value or values of c that satisfy the equation (f(b) − f(a)) / (b − a) = f′(c) in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–6.


g(x) = {x³, −2 ≤ x ≤ 0

x², 0 < x ≤ 2

Textbook Question

37. What value of a makes f(x) = x^2 +(a/x) have

a. a local minimum at x = 2?

b. a point of inflection at x = 1?