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)?

Solution Steps

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

• \( f(5) = 3 \)
• \( f'(5) = 3 \)
• \( f''(5) = -2 \)

• \( f(5) = 3 \): 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.

• 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