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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Multiple Choice

Suppose the graph of $y = f (x)$ is shown above. Which of the following best describes the graph of $y = f^{″} (x)$ ?

It is zero wherever the graph of

y = f (x)

has a local maximum or minimum.

It is positive where the graph of

y = f (x)

is increasing and negative where

y = f (x)

is decreasing.

It is always positive if

y = f (x)

is above the

x

-axis.

It is positive where the graph of

y = f (x)

is concave up and negative where

y = f (x)

is concave down.

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Verified step by step guidance

Step 1: Understand the relationship between the function y = f(x), its first derivative y = f'(x), and its second derivative y = f''(x). The second derivative provides information about the concavity of the graph of y = f(x).

Step 2: Recall that concavity describes how the graph curves. If y = f''(x) > 0, the graph of y = f(x) is concave up (curving upwards like a U). If y = f''(x) < 0, the graph of y = f(x) is concave down (curving downwards like an upside-down U).

Step 3: Identify the points of inflection on the graph of y = f(x). These are the points where the concavity changes, and y = f''(x) is zero at these points.

Step 4: Analyze the behavior of y = f(x) in regions where it is concave up or concave down. In regions where y = f(x) is concave up, y = f''(x) will be positive. In regions where y = f(x) is concave down, y = f''(x) will be negative.

Step 5: Conclude that the correct description of the graph of y = f''(x) is: 'It is positive where the graph of y = f(x) is concave up and negative where y = f(x) is concave down.' This aligns with the properties of the second derivative.