Math Problem Statement

Solution

To determine which of the statements are true based on the graph of ff':

  1. Relative Min at x=3x = -3:

    • A relative minimum of ff occurs where ff' changes from negative to positive.
    • At x=3x = -3, ff' is increasing (it changes from negative to positive).
  2. Point of Inflection at x=2x = 2:

    • A point of inflection of ff occurs where ff'' (the second derivative) changes sign, which corresponds to ff' having a local extremum (either a max or min).
    • At x=2x = 2, ff' has a local minimum, indicating that ff'' changes sign, so there is a point of inflection at x=2x = 2.
  3. Concave Down for 0<x<40 < x < 4:

    • ff is concave down where f<0f'' < 0, which corresponds to ff' being decreasing.
    • For 0<x<40 < x < 4, ff' is indeed decreasing, indicating ff is concave down in this interval.

Based on the above analysis, all three statements are true:

  • I is true because ff has a relative minimum at x=3x = -3.
  • II is true because the graph of ff has a point of inflection at x=2x = 2.
  • III is true because the graph of ff is concave down for 0<x<40 < x < 4.

Therefore, the correct answer is:

All three statements (I, II, and III) are true.

Would you like more details or have any questions?

Here are some questions you might ask next:

  1. Can you explain how to identify points of inflection on a graph of ff'?
  2. How do you determine concavity from the graph of ff'?
  3. What are the key features to look for on the graph of a derivative?
  4. Can you give an example of how to find relative extrema using the first derivative?
  5. How does the behavior of ff' relate to the concavity of ff?

Tip: When analyzing the graph of a derivative, remember that points where the derivative changes sign can indicate relative extrema of the original function.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Inflection Points
Concavity

Formulas

First derivative test
Second derivative test

Theorems

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Suitable Grade Level

Advanced High School and College