Let $D$ denote the unit disk centered at the origin. I'm trying to find the maximum and minimum values of the function $$u:D \to \mathbb{R},\,\ (x,y) \mapsto e^x \cos y.$$
I'll try using Lagrange multipliers:
Set $g(x,y)-1 := x^2+y^2-1$ and write $$\mathcal{L}(x,y, \lambda)=u(x,y)+ \lambda(g(x,y)-1) = e^x \cos y + \lambda(x^2+y^2-1).$$ Then $\nabla \mathcal{L}(x,y, \lambda)= (\mathcal{L}_x,\mathcal{L}_y,\mathcal{L}_{\lambda})=(e^x \cos y + 2 \lambda x, -e^x \sin y + 2 \lambda y, x^2+y^2-1)$. Thus $$\nabla \mathcal{L}(x,y,\lambda) =0 \iff \begin{cases} e^x \cos y + 2 \lambda x = 0 \\ -e^x \sin y + 2 \lambda y = 0 \\ x^2+y^2-1=0 \end{cases}$$
Here's where I'm having trouble. I'm trying to solve this system of equations for $\lambda$, so that I can plug the points back into $u(x,y)$, but I can't seem to isolate any of the variables. Any help appreciated!
$\cos^2 x+\sin^2 x=1$ is usually a powerful formula. Since \begin{align} (2\lambda x)^2+(2\lambda y)^2 &= 4\lambda^2\\ &=e^{2x}, \end{align}
\begin{align} -e^x \sin y + 2\lambda y &= e^x (\pm y-\sin y). \end{align} Since $e^x>0$, $\pm y -\sin y =0$ and $y=0$ is the unique root. Therefore, given function has two extreme point, $(1,0)$ and $(-1,0)$ and $e$ and $1/e$ are maximum and minimum each.