Questions: Problem 4 The graph of a function f on the interval (-6,7) is shown in the figure below. (a) On what interval(s) is f decreasing? (b) On what interval(s) is f increasing? (c) On what interval(s) is f concave up? (d) On what interval(s) is f concave down?

Problem 4 The graph of a function f on the interval (-6,7) is shown in the figure below. (a) On what interval(s) is f decreasing? (b) On what interval(s) is f increasing? (c) On what interval(s) is f concave up? (d) On what interval(s) is f concave down?
Transcript text: Problem 4 The graph of a function $f$ on the interval $(-6,7)$ is shown in the figure below. (a) On what interval(s) is $f$ decreasing? (b) On what interval(s) is $f$ increasing? (c) On what interval(s) is $f$ concave up? (d) On what interval(s) is $f$ concave down?
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Solution

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Solution Steps

Step 1: Identify the intervals where the function is decreasing

The function \( f(x) \) is decreasing where the graph is moving downwards as \( x \) increases. From the graph, this occurs in the intervals:

  • \( (-6, -2) \)
  • \( (3, 5) \)
  • \( (6, 7) \)
Step 2: Identify the intervals where the function is increasing

The function \( f(x) \) is increasing where the graph is moving upwards as \( x \) increases. From the graph, this occurs in the intervals:

  • \( (-2, 3) \)
  • \( (5, 6) \)
Step 3: Identify the intervals where the function is concave up

The function \( f(x) \) is concave up where the graph is curving upwards. From the graph, this occurs in the interval:

  • \( (4, 5) \)

Final Answer

(a) The function \( f(x) \) is decreasing on the intervals \( (-6, -2) \), \( (3, 5) \), and \( (6, 7) \).

(b) The function \( f(x) \) is increasing on the intervals \( (-2, 3) \) and \( (5, 6) \).

(c) The function \( f(x) \) is concave up on the interval \( (4, 5) \).

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