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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2:47 minutes

Problem 4.4.117

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?

Verified step by step guidance

To find the x-values where the graph of f has an inflection point, we need to determine where the second derivative changes sign. An inflection point occurs where the concavity of the function changes.

Start by setting the second derivative equal to zero:

y'' = (x+1)(x-2) = 0

. Solve this equation to find the critical points.

Factor the equation:

(x+1)(x-2) = 0

. This gives us two solutions:

x = -1

and

x = 2

To confirm these are inflection points, check the sign of the second derivative around these values. Choose test points in the intervals

(-	ext{∞}, -1)

(-1, 2)

, and

(2, 	ext{∞})

Evaluate the sign of

y''

at these test points to determine if the sign changes. If the sign changes, then the corresponding x-value is an inflection point.

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Key Concepts

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

Second Derivative Test

The second derivative test is used to determine the concavity of a function and identify inflection points. An inflection point occurs where the second derivative changes sign, indicating a transition from concave up to concave down or vice versa. For the function y = f(x), the inflection points are found by setting the second derivative equal to zero and solving for x.

Factoring Quadratic Expressions

Factoring quadratic expressions involves rewriting them as a product of linear factors. In the given second derivative y'' = (x+1)(x-2), the expression is already factored, indicating potential x-values where the derivative equals zero. These values, x = -1 and x = 2, are critical for determining where the graph might have inflection points.

Sign Change Analysis

Sign change analysis involves examining the intervals around the roots of the second derivative to determine where the sign changes. This helps confirm the presence of inflection points. For y'' = (x+1)(x-2), check the sign of y'' in intervals around x = -1 and x = 2 to ensure it changes, confirming these x-values as inflection points.

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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

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.

Textbook Question

114. Parabolas

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

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.

Textbook Question

103. A function f(x) has domain (-2, 2). The graph below is a plot of the derivative of f, not a plot of f itself. In other words, this is a graph of y = f'(x). Either use this graph to determine on which intervals the graph of f is concave up and on which intervals the graph of f is concave down, or explain why this information cannot be determined from the graph.

Textbook Question

107. Marginal cost The accompanying graph shows the hypothetical cost c=f(x) of manufacturing x items. At approximately what production level does the marginal cost change from decreasing to increasing?

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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?

Key Concepts

Second Derivative Test

Factoring Quadratic Expressions

Sign Change Analysis

Watch next

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?