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

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Answer 1 · 2016-11-29T23:20:32

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

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 $x=-1$ .

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