How do you solve the Lagrange multiplier system for the objective function $f(x,y) = x^2 + y^2$ and the constraint $g(x,y) = x^2 + y^2/2 - 1 = 0$?

Conditions

• The objective function is $f(x,y) = x^2 + y^2$.
• The constraint function is $g(x,y) = x^2 + y^2/2 - 1 = 0$.
• The Lagrange multiplier condition $\nabla f = \lambda \nabla g$ is applied.

Reasoning, step by step

1. Calculate the gradients: $\nabla f = \langle 2x, 2y \rangle$ and $\nabla g = \langle 2x, y \rangle$.
2. Set up the component equations: $2x = \lambda(2x)$ and $2y = \lambda(y)$.
3. Include the constraint equation: $x^2 + y^2/2 - 1 = 0$.
4. Solve the system of three equations for $x$, $y$, and $\lambda$.
5. Identify the resulting critical points: $(0, \sqrt{2})$, $(0, -\sqrt{2})$, $(1, 0)$, and $(-1, 0)$.

Example

The video explicitly shows the equations $2x = \lambda(2x)$ and $2y = \lambda(y)$ combined with the constraint. It then states that solving this system yields the four points $(0, \sqrt{2})$, $(0, -\sqrt{2})$, $(1, 0)$, and $(-1, 0)$.

Common misconceptions

• Forgetting to include the original constraint equation in the system.
• Assuming $\lambda$ must be positive, when it can be negative or zero depending on the geometry.

Watch the explanation

Connected concepts

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