Question

Find the extreme values of the function f(x, y) = 4x + 8y subject to the constraint g(x, y) = x2 + y2 − 5 = 0.

Answer 1

`L(x,y,\lambda)=4x+8y+\lambda(x^2+y^2-5)`


`L_x=4+2\lambda x=0`

`L_y=8+2\lambda y=0`

`L_\lambda=x^2+y^2-5=0`


`L_y-2L_x=2\lambda y-4\lambda x=2\lambda(y-2x)=0\implies y=2x`


`xL_x+yL_y=2\lambda(x^2+y^2)+4x+8y=10\lambda+4x+8y=0\iff5\lambda+2x+4y=0`


`y=2x\implies 5\lambda+5y=0\implies 5(\lambda+y)=0\implies y=-\lambda`


`4+2\lambda x=0\iff4+\lambda y=0\iff4-y^2=0\implies y=\pm2`

`y=\pm2\implies x=\frac y2=\pm1`

In other words, there are two critical points,
`(1,2)` and
`(-1,-2)`. At the first point, there is a maximum of
`f(1,2)=20` and at the second, there is a minimum of
`f(-1,-2)=-20`.