Using Lagrange multipliers,
$$\begin{array}{ll} \text{extremize} & x + y\\ \text{subject to} & x^{2} + y^{2} \leq 5\\ & x \geq 0\end{array}$$
Graphically, it comes that the maximiser is $(\sqrt{5}, 0)$, but the minimizer is unclear. How to find it? How should Lagrange multiplier looks like?
For this simple problem you don't need Lagrange multipliers. Instead draw a figure.
The feasible domain $B$ is a right half disc of radius $\sqrt{5}$. The objective function has level lines $x+y=c$, which are lines descending with slope $45^\circ$. When $c$ gets larger the line is translated in direction north-east. Now find out which is the most south-west such line meeting a point of $B$, and which is the most north-east such line meeting a point of $B$.
Note that one of the two points would not be found using Lagrange multipliers.