Questions: In each part, list the points (A-F) on the graph of f whose x-coordinates satisfy the given conditions. (a) f'(x) > 0 and f''(x) > 0 (b) f'(x) < 0 and f''(x) > 0 (c) f'(x) = 0 and f''(x) < 0 (d) f'(x) = 0 and f''(x) > 0 (e) f'(x) < 0 and f''(x) = 0 (a) Select all that apply. Point C Point F Point B Point D Point E Point A

In each part, list the points (A-F) on the graph of f whose x-coordinates satisfy the given conditions. (a) f'(x) > 0 and f''(x) > 0 (b) f'(x) < 0 and f''(x) > 0 (c) f'(x) = 0 and f''(x) < 0 (d) f'(x) = 0 and f''(x) > 0 (e) f'(x) < 0 and f''(x) = 0 (a) Select all that apply. Point C Point F Point B Point D Point E Point A
Transcript text: In each part, list the points (A-F) on the graph of $f$ whose $x$-coordinates satisfy the given conditions. (a) $f^{\prime}(x)>0$ and $f^{\prime \prime}(x)>0$ (b) $f^{\prime}(x)<0$ and $f^{\prime \prime}(x)>0$ (c) $f^{\prime}(x)=0$ and $f^{\prime \prime}(x)<0$ (d) $f^{\prime}(x)=0$ and $f^{\prime \prime}(x)>0$ (e) $f^{\prime}(x)<0$ and $f^{\prime \prime}(x)=0$ (a) Select all that apply. Point $C$ Point F Point B Point D Point E Point A
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Solution

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Solution Steps

Step 1: Understanding the Problem

We need to identify points on the graph where the first derivative \( f'(x) \) and the second derivative \( f''(x) \) satisfy given conditions.

Step 2: Analyzing the Graph
  • \( f'(x) \) represents the slope of the function.
  • \( f''(x) \) represents the concavity of the function.
Step 3: Solving Part (a)
Condition: \( f'(x) > 0 \) and \( f''(x) > 0 \)
  • \( f'(x) > 0 \): The function is increasing.
  • \( f''(x) > 0 \): The function is concave up.

From the graph:

  • Point E: The function is increasing and concave up.
  • Point F: The function is increasing and concave up.

Final Answer

  • Point E
  • Point F
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