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

Since

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

First finding the second derivative:

We see that

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To determine if ( f(x) = \sin(x) ) is concave or convex at ( x = -1 ), we need to find the second derivative of ( f(x) ) and evaluate it at ( x = -1 ). The second derivative will tell us about the concavity of the function.

The first derivative of ( f(x) = \sin(x) ) is ( f'(x) = \cos(x) ).

The second derivative of ( f(x) ) is ( f''(x) = -\sin(x) ).

When ( x = -1 ), ( f''(-1) = -\sin(-1) ).

Since ( \sin(-1) ) is positive, ( f''(-1) ) is negative.

A function is concave where its second derivative is negative, so at ( x = -1 ), ( f(x) = \sin(x) ) is concave.

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