Concavity of a Function

==2024-12-14

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Don’t underestimate the power of a cavity.

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Theory

Concavity of a function $f(x)$ is the curvature of the function at the given interval. In the figure, you can see that when $f^{\prime \prime }(x)<0$, there is Concave Downwards, and when $f^{\prime \prime }(x)>0$, there is Concave Upwards.

Point of Inflection

The point where the concavity of the graph changes is the point of inflection.

We should not always rely on the concept that when $f^{\prime \prime }(x)=0$, there will be point of inflection.

But where $2^{\text{nd}}$ derivative might not be defined, there might still be a point of inflection. It is just the matter of intervals and what $f^{\prime \prime }(x)$ compares with $0$ in that intervals.

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PTR