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.
Table of contents
- 0. Functions7h 52m
- Introduction to Functions16m
- Piecewise Functions10m
- Properties of Functions9m
- Common Functions1h 8m
- Transformations5m
- Combining Functions27m
- Exponent rules32m
- Exponential Functions28m
- Logarithmic Functions24m
- Properties of Logarithms34m
- Exponential & Logarithmic Equations35m
- Introduction to Trigonometric Functions38m
- Graphs of Trigonometric Functions44m
- Trigonometric Identities47m
- Inverse Trigonometric Functions48m
- 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
- 10. Physics Applications of Integrals 3h 16m
- 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
- 16. Parametric Equations & Polar Coordinates7h 58m
5. Graphical Applications of Derivatives
Concavity
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For the following graph, find the open intervals for which the function is concave up or concave down. Identify any inflection points.

A
Concave up: (−∞,0), (1,∞); Concave down: (0,1); Inflection points: (0,0), (1,1)
B
Concave up: (−∞,0); Concave down: (0,∞); Inflection points: (0,0)
C
Concave down: (−1,0), (1,∞); Concave up: (0,1); Inflection points: (0,0), (1,1)
D
Concave up: (−1,0), (1,∞); Concave down: (0,1); Inflection points: (−1,1), (0,0), (1,1)

1
To determine where the function is concave up or down, we need to analyze the second derivative of the function. The second derivative tells us about the concavity of the function.
Identify the points where the concavity changes, known as inflection points. These occur where the second derivative is zero or undefined, and the concavity changes sign.
From the graph, observe the intervals where the curve is bending upwards (concave up) and downwards (concave down). The curve is concave up when it is shaped like a cup (U) and concave down when it is shaped like a cap (∩).
In the graph, the function appears to be concave down on the interval (0, 1) and concave up on the intervals (-∞, 0) and (1, ∞).
The inflection points, where the concavity changes, are at the points (0, 0) and (1, 1) as these are the points where the curve changes from concave down to concave up or vice versa.
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