Question

Is `f(x)=sinx/cosx` concave or convex at `x=0`?

Answer 1

It is at a point of inflection.

Explanation:

The function is `f(x)=sinx/cosx=tanx`.

To find out whether it is concave or convex at `x-0`, let us find average of the value of `f(x)` at `x=0+Deltax=Deltax` and `x=0-Deltax=-Deltax`.

Ax `tan(-Deltax)=-tanDeltax`, this should be `0`.

Also `tanx=0` at `x=0`.

Hence, at `x=0`, `f(x)=sinx/cosx=tanx` is neither concave nor convex.

If average of `f(x_0+Deltax))` and `f(x_0-Deltax))` is less than that at `f(x_0)`, it is concave and if average of `f(x_0+Deltax))` and `f(x_0-Deltax))` is less than that at `f(x_0)`, it is convex. If they are equal it is point of inflection.

It is at a point of inflection.