Concavity Functions and the Second Derivative test

[mathgeek22]

Hi,
The question is find the critical points of f(x)=sin^2(x)+cos(x) in the interval (-pi,pi) and apply the Second Derivative test is possible to determine whether each corresponds to a local minimum or maximum..

My question is when we are figuring out the second derivative do we set that equal to 0 or -pi &pi. Because I do notice that -pi,pi are not included values.

Please let me know.
Thanks

 

[tkhunny]

Hi,
The question is find the critical points of f(x)=sin^2(x)+cos(x) in the interval (-pi,pi) and apply the Second Derivative test is possible to determine whether each corresponds to a local minimum or maximum..

My question is when we are figuring out the second derivative do we set that equal to 0 or -pi &pi. Because I do notice that -pi,pi are not included values.

Please let me know.
Thanks

This is good question - very important to ponder.

The Second-Derivative test for Critical Points is \(\displaystyle f^{[2]}(a_{i}) = 0,\;for\;L < a_{i} < H\), where \(\displaystyle L\;and\;H\) are your specified or inherent limits. In this case, you have \(\displaystyle L = -\pi\;and\;H = \pi\). Your task is to find all the "a" values - as many as exist. It is VERY important to notice that these are strict inequalities. The derivative is unlikely to exist AT \(\displaystyle L\;or\;H\). You must investigate these separately.