# Thread: First derivative test, local min/max

1. ## First derivative test, local min/max

Hi

I have a function

$g(x)=(cos \ x + sin \ x - 1)e^{cos x + sin x} \ \ \ \ \ (0 \leq x \leq\pi)$

the derivative being

$g`(x)=(cos^2 \ x - sin^2 \ x)e^{cos x + sin x}$

I'm not sure how to find the turning point within this domain.

Any ideas?

2. $g'(x)=\left( \cos ^{2}x-\sin ^{2}x \right)e^{\cos x+\sin x}=0,$ and this is just $\cos ^{2}x-\sin ^{2}x=0$ since $e^{\cos x+\sin x}>0$ for each $x\in\mathbb R.$ Now find solutions for $0\le x\le\pi$ for that trig. equation.

3. This could simply things.
$\cos ^2 (x) - \sin ^2 (x) = \cos (2x)$