Find the critical points of the function $$z=f(x,y)=x^{2}+y^{2}-2x+4y$$

Solution  The partial derivatives of $f$ are $$f_{x}=2x-2\qquad \hbox{and} \qquad f_{y}=2y+4$$

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Since both partial derivatives exist for all $x$ and $y$, the critical points of $f$ satisfy the system of equations $$\left\{\begin{array}{l} 2x-2=0\\[3pt] 2y+4=0 \end{array}\right.$$

Solving the system simultaneously, we find that the only critical point is $(1,-2)$.