Finding Critical Points

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 \( (1,-2) \).