Question

Use the method of lagrange multipliers to minimize the function subject to the given constraint. (round your answers to three decimal places.) minimize the function f(x, y) = x2 + 3y2 subject to the constraint x + y ? 1 = 0.

Answer 1

The Lagrangian for this problem is

$L(x,y,\lambda)=x^2+3y^2+\lambda(x+y-1)$

and has partial derivatives

$\begin{cases}L_x=2x+\lambda\\L_y=6y+\lambda\\L_\lambda=x+y-1\end{cases}$

Set each partial derivative equal to 0 and solve for $x$ and $y$:

$\begin{cases}2x+\lambda=0\\6y+\lambda=0\\x+y=1\end{cases}$

Subtracting the second equation from the first, we get

$2x-6y=0\implies x-3y=0$

and subtracting this from the third equation yields

$4y=1\implies y=\dfrac14$

which means

$x+\dfrac14=1\implies x=\dfrac34$

So a critical point occurs at $\left(\dfrac34,\dfrac14\right)$ (or (0.750, 0.250)). The minimum value would then be $f\left(\dfrac34,\dfrac14\right)=\dfrac34=0.750$.