What is the concavity of a linear function?

The concavity of a linear function is zero.

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Concavity measures the rate at which the slope of a graph increases (“concave up”) or decreases (“concave down”). Since the slope of a linear function is constant, it is neither increasing nor decreasing at all, so its concavity is zero!

In terms of derivatives, the concavity of a graph at a point corresponds to the second derivative of the function at that point. For example, the linear function ##f(x)=3x-5## has derivative ##(df)/(dx)=3##. The second derivative is ##(d^2f)/(dx^2)=0## and so the concavity is zero.

For comparison:

##f(x)=x^2## has derivative ##(df)/(dx)=2x## and second derivative ##(d^2f)/(dx^2)=2##. Its concavity is 2 (everywhere) which makes sense since the graph is a concave up parabola.

##f(x)=x^3## has derivative ##(df)/(dx)=3x^2## and second derivative ##(d^2f)/(dx^2)=6x##. We see that the concavity is sometimes positive (when ##x>0##) and sometimes negative (when ##x<0##). Also, there is a point of inflection at ##x=0## and the concavity is 0 there.

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