Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

5. Graphical Applications of Derivatives

Concavity

Calculus

5. Graphical Applications of Derivatives

Concavity

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3:14 minutes

Problem 4.4.113

Textbook Question

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

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.

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.

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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Related Practice

Textbook Question

The graph of f' on the interval [-3,2] is shown in the figure. <IMAGE>

c. At what point(s) does f have an inflection point?

Textbook Question

Does ƒ(x) = (x⁶/2) + (5x⁴/4) - 15x² have any inflection points? If so, identify them.

Textbook Question

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>

e. On what intervals (approximately) is f concave up?

Textbook Question

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>

f. On what intervals (approximately) is f concave down?

Textbook Question

114. Parabolas

b. When is the parabola concave up? Concave down?

Textbook Question

117. Suppose that the second derivative of the function y = f(x) isy" =(x+1)(x-2).

For what x-values does the graph of f have an inflection point?

Textbook Question

Graph f(x) = x cos x and its second derivative together for 0 ≤ x ≤ 2pi. Comment on the behavior of the graph of f in relation to the signs and values of f".

Textbook Question

Each of Exercises 89–92 shows the graphs of the first and second derivatives of a function y=f(x). Copy the picture and add to it a sketch of the approximate graph of f, given that the graph passes through the point P.

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

Key Concepts

Point of Inflection

Second Derivative

Polynomial Derivatives

Watch next

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.