228k views
0 votes
Use the method of lagrange multipliers to find

a. the minimum value of x + y​, subject to the constraints xy=64, x>0, y>0

User ChangUZ
by
8.5k points

1 Answer

1 vote
The Lagrangian,


L(x,y,\lambda)=x+y+\lambda(xy-64)

has partial derivatives (set equal to 0)


L_x=1+\lambda y=0\implies\lambda=-\frac1y

L_y=1+\lambda x=0\implies\lambda=-\frac1x

L_\lambda=xy-64=0

The first two equations tell us that
-\frac1y=-\frac1x\implies x=y.

Substituting this into the second equation, we have


xy-64=0\iff x^2=0\implies x=\pm8\implies\begin{cases}x=8,y=8\\x=-8,y=-8\end{cases}

We have that
x,y>0, so we only have the one critical point on the surface
x+y at (8, 8), with an extreme value of 16.
User Keith Gaddis
by
8.7k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories