Geometry of Functions X: Determining Concavity From Second Derivatives
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We have defined concavity as the increase or decrease of a function's derivative. If a function's derivative is increasing on an interval then the function is concave up. If it's derivative is decreasing then the function is concave down.
Recall that a derivative is a rate of change. It expresses how quickly something is increasing or decreasing. The derivative of a function's first derivative is going to express how quickly the first derivative is increasing or decreasing. But the derivative of a function's first derivative is the function's second derivative. Therefore, we can use the second derivative to determine concavity.
For example, let's suppose that we have a function where it is concave up on some domain. Since it is concave up it means that as we go from left to right on the graph the slope is increasing. If $u$ and $v$ are in the domain where the function is concave up and <Sign in to see all the formulas> then
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If <Sign in to see all the formulas> then
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because <Sign in to see all the formulas> at the same time that <Sign in to see all the formulas>. This means that
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But we know that
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Therefore, in the domain where a function is concave up
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It's second derivative is greater than zero!
Similar logic shows that for a concave down function that
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For example, we have already shown that <Sign in to see all the formulas> is concave up everywhere. Will the second derivative show us this as well? We know that <Sign in to see all the formulas>. Therefore, because the second derivative is greater than zero we would have quickly and easily seen that $x^2$ is concave up.
Another example is the concavity of <Sign in to see all the formulas>. We know that <Sign in to see all the formulas>. When <Sign in to see all the formulas> the second derivative is less than zero which corresponds to the domain where $x^3$ is concave down. When <Sign in to see all the formulas> the second derivative is greater than zero which corresponds to the domain where it is concave up. The picture below demonstrates this and is the best picture to keep in your head to remember how concavity and second derivatives are related.
