What is the concavity of a linear function?

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The flatness of a rectirectistraight administration is cipher.

Concavity measures the objurgate at which the extend of a graph increases ("excavated up") or decreases ("excavated down"). Since the extend of a rectirectistraight administration is faithful, it is neither increasing nor decreasing at all, so its flatness is cipher!

In provisions of derivatives, the flatness of a graph at a purpose corresponds to the succor derivative of the administration at that purpose. For illustration, the rectirectistraight administration ##f(x)=3x-5## has derivative ##(df)/(dx)=3##. The succor derivative is ##(d^2f)/(dx^2)=0## and so the flatness is cipher.

For comparison:

##f(x)=x^2## has derivative ##(df)/(dx)=2x## and succor derivative ##(d^2f)/(dx^2)=2##. Its flatness is 2 (everywhere) which makes import since the graph is a excavated up parabola.

##f(x)=x^3## has derivative ##(df)/(dx)=3x^2## and succor derivative ##(d^2f)/(dx^2)=6x##. We see that the flatness is casually overbearing (when ##x>0##) and casually denying (when ##x<0##). Also, there is a purpose of inflexion at ##x=0## and the flatness is 0 there.

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