Find, if existent, the global extrema of $\varphi:\mathbb{R}^3\to\mathbb{R},\, (x,y,z)\mapsto \exp(x+y+z)$ on $M:= \left\{ \begin{pmatrix}x\\y\\z\end{pmatrix}\in\mathbb{R}^3 : \exp(x)+y^2+z^2=1 \right\}$.
Since $M$ is compact, we should be able to find both, global maximum and global minimum on $M$. However, my computation yields only one critical point.
My computation:
Set $\Phi(x,y,z,\lambda):=\varphi(x,y,z)-\lambda\cdot (\exp(x)+y^2+z^2-1)$. Letting $D\Phi=0$ gives the following equations:
$\exp(x,y,z)=\lambda\exp(x)=2\lambda y=2\lambda z$ and $\exp(x)+y^2+z^2=1$ which results in $x=\ln(2y),\, y=z$ and ultimately gives us $y_{1,2}=-\frac{1}{2}\pm\sqrt{\frac{3}{4}}$, of which we can only use the positive solution since $x=\ln(2y)$. Where have I made a mistake?