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?
Table of contents
- 0. Functions7h 55m
- Introduction to Functions18m
- Piecewise Functions10m
- Properties of Functions9m
- Common Functions1h 8m
- Transformations5m
- Combining Functions27m
- Exponent rules32m
- Exponential Functions28m
- Logarithmic Functions24m
- Properties of Logarithms36m
- 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 31m
- 12. Techniques of Integration7h 41m
- 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
Multiple Choice
Suppose the graph of is shown above. Which of the following best describes the graph of ?
A
It is zero wherever the graph of has a local maximum or minimum.
B
It is positive where the graph of is increasing and negative where is decreasing.
C
It is always positive if is above the -axis.
D
It is positive where the graph of is concave up and negative where is concave down.
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Verified step by step guidance1
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.
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