The curvature of a function shows how the function bends. Where the function has a maximum, it bends downwards, and we call it concave. Where the function has a minimum, the graph bends upwards, and we call it convex.
You find the curvature of a graph by looking at the sign chart of the second derivative
Rule
The following relationship exists between the second derivative and the curvature of the graph.
If the second derivative of the function is positive, the graph is positive (looks like it’s smiling). If the second derivative of the function is negative, the graph is negative (looks like it’s sad).
Example 1
Describe the curvature of the graph given by
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First, you differentiate the function twice like this:
By putting
As
It’s good to remember that
Rule
You have a solid line in the sign chart when:
You have a dotted line when: