Questions: The graph of y=f(x) has f(5)=3, f'(5)=3, and f''(5)=-2. Which of the following might be a graph of y=f(x)?

The graph of y=f(x) has f(5)=3, f'(5)=3, and f''(5)=-2. Which of the following might be a graph of y=f(x)?
Transcript text: The graph of $y=f(x)$ has $f(5)=3, f^{\prime}(5)=3$, and $f^{\prime \prime}(5)=-2$. Which of the following might be a graph of $y=f(x)$ ?
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Solution

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Solution Steps

Step 1: Identify the given conditions

The problem states that the graph of \( y = f(x) \) has the following properties at \( x = 5 \):

  • \( f(5) = 3 \)
  • \( f'(5) = 3 \)
  • \( f''(5) = -2 \)
Step 2: Analyze the conditions
  • \( f(5) = 3 \): The point (5, 3) must be on the graph.
  • \( f'(5) = 3 \): The slope of the tangent line at \( x = 5 \) is 3, indicating an upward slope.
  • \( f''(5) = -2 \): The concavity at \( x = 5 \) is negative, indicating the graph is concave down at this point.
Step 3: Examine the graphs
  • The graph must pass through the point (5, 3).
  • The graph must have a positive slope at \( x = 5 \).
  • The graph must be concave down at \( x = 5 \).

Final Answer

The correct graph is the one in the top left corner. This graph passes through the point (5, 3), has a positive slope at \( x = 5 \), and is concave down at \( x = 5 \).

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