Mathematics 241 – Multivariable Calculus Solution for Quiz 7 – November 9, 2007

(10) Find the critical points of the function f(x, y) = 3x+2y on the constraint set g(x, y) = x^{2} + y^{2} − 4 = 0 by the method of Lagrange multipliers.

Solution: The Lagrange multiplier equations are

\begin{aligned}\frac{∂f}{∂x} & = λ\frac{∂g}{∂x} \\ \frac{∂f}{∂y} & = λ\frac{∂g}{∂y} \\ g(x, y) & = 0 \end{aligned}

or

\begin{aligned}3 & = 2λx \\ 2 & = 2λy \\ x^{2} + y^{2} − 4 & = 0. \end{aligned}

From the first two equations,

3y = 2λxy = 2x,

so we can eliminate λ to obtain:

y = \frac{2x}{3}.

Substitute this into the constraint equation to eliminate y:

x^{2} + (\frac{2x}{3})^{2} = x^{2} + \frac{4x^{2}}{9} = \frac{13x^{2}}{9} = 4.

This says x^{2} = \frac{36}{13}, so x = ±\frac{6}{\sqrt{13}}. Then from the equation y = \frac{2x}{3}, we get the corresponding values of y, y = ±\frac{4}{\sqrt{13}}. Note that y must have the same sign as x, so there are only two critical points here:

(\frac{6}{\sqrt{13}}, \frac{4}{\sqrt{13}}) , (−\frac{6}{\sqrt{13}}, −\frac{4}{\sqrt{13}}) .