Question

Is `f(x)=sinx` concave or convex at `x=-1`?

Answer 1

Since `f''(-1)>0`, we see that `sinx` is convex ("concave up") at `x=-1`.

Explanation:

We have to know that `d/dxsinx=cosx` and `d/dxcosx=-sinx`.

Also recall that concavity and convexity are determined through the sign of the second derivative of a function.

First finding the second derivative:

`f(x)=sinx`

`f'(x)=cosx`

`f''(x)=-sinx`

If `f''(-1)>0`, then `f` is convex (commonly called "concave up") at `x=-1`.

If `f''(-1)<0`, then `f` is concave (commonly called "concave down") at `x=-1`.

We see that

`f''(-1)=-sin(-1)approx08415`

Since `f''(-1)>0`, we see that `sinx` is convex ("concave up") at `x=-1`.