Answer to Question #107930 in Calculus for annie

Question

Use Lagrange multipliers to find the maximum and minimum values of f(x,y)=x−2y subject to the constraint x2+y2=5, if such values exist.

maximum = ?

minimum = ?

Answer 1

$\begin{aligned} & f(x,y)=x-2y \\ & g(x,y)={{x}^{2}}+{{y}^{2}}=5 \\ & \text{Use Lagrange multipliers,} \\ & \left\langle {{f}_{x}},{{f}_{y}} \right\rangle =\lambda \left\langle {{g}_{x}},{{g}_{y}} \right\rangle \\ & \left\langle 1,-2 \right\rangle =\lambda \left\langle 2x,2y \right\rangle \\ & \text{Compare both sides} \\ & \frac{1}{2x}=\frac{-2}{2y}\Rightarrow 2y=-4x\Rightarrow y=-2x \\ & \text{Substitute this in }{{x}^{2}}+{{y}^{2}}=5 \\ & {{x}^{2}}+{{\left( -2x \right)}^{2}}=5\Rightarrow 5{{x}^{2}}=5\Rightarrow x=\pm \sqrt{1}\Rightarrow x=\pm 1 \\ & \text{Substitute this in }y=-2x \\ & \text{When }x=1,y=-2\left( 1 \right)=-2 \\ & \text{When }x=-1,y=-2\left( -1 \right)=2 \\ & f(1,-2)=1-2(-2)={\color{red}{5}}\leftarrow maximum~ \\ & f(-1,2)=-1-2(2)={\color{red}{-5}}\leftarrow minimum~ \\ \end{aligned}$