The second derivative provides information about which property of a function?

The second derivative provides information about which property of a function?

• A. the average rate of change
• B. the value of the function
• C. the definite integral
• D. concavity of the function

Answer: (B)

The second derivative tells you about the graph’s concavity and how the slope is changing. The first derivative f′(x) gives the slope of the tangent line, so it describes how fast the function is increasing or decreasing at a point. The second derivative f″(x) looks at how that slope itself behaves as x changes: if f″(x) > 0, the slope is increasing and the graph bends upward (concave up); if f″(x) < 0, the slope is decreasing and the graph bends downward (concave down). When f″(x) = 0, you might have a point of inflection where the concavity changes, though that depends on the surrounding sign of f″(x). This is not about the function’s value at a point, the total accumulated area, or the average rate of change over an interval—the second derivative specifically captures the curvature/concavity of the graph.

The second derivative tells you about the graph’s concavity and how the slope is changing. The first derivative f′(x) gives the slope of the tangent line, so it describes how fast the function is increasing or decreasing at a point. The second derivative f″(x) looks at how that slope itself behaves as x changes: if f″(x) > 0, the slope is increasing and the graph bends upward (concave up); if f″(x) < 0, the slope is decreasing and the graph bends downward (concave down). When f″(x) = 0, you might have a point of inflection where the concavity changes, though that depends on the surrounding sign of f″(x). This is not about the function’s value at a point, the total accumulated area, or the average rate of change over an interval—the second derivative specifically captures the curvature/concavity of the graph.