If a function's second derivative is negative, what does it imply about the graph of the function?

If a function's second derivative is negative, what does it imply about the graph of the function?

• A. The graph is concave up
• B. The graph is concave down
• C. The graph crosses the x-axis
• D. The graph has no slope

Answer: (B)

When the second derivative of a function is negative, it indicates that the graph of the function is concave down. This means that as you move from left to right along the graph, the slope of the tangent line decreases, which leads to the shape curving downwards like an upside-down bowl. In practical terms, this characteristic of the graph implies that if there are critical points on the graph (where the first derivative is zero), those points would represent local maxima rather than minima. This is significant in understanding the behavior of the function and predicting the nature of its increasing and decreasing intervals.

When the second derivative of a function is negative, it indicates that the graph of the function is concave down. This means that as you move from left to right along the graph, the slope of the tangent line decreases, which leads to the shape curving downwards like an upside-down bowl.

In practical terms, this characteristic of the graph implies that if there are critical points on the graph (where the first derivative is zero), those points would represent local maxima rather than minima. This is significant in understanding the behavior of the function and predicting the nature of its increasing and decreasing intervals.