Question

Find the minimum and maximum values of the function subject to the given constraint. (if an answer does not exist, enter dne.) f(x, y) = x2y + x + y, xy = 5

Answer 1

Via Lagrange multipliers:


`L(x,y,\lambd)=x^2y+x+y+\lambda(xy-5)`

`L_x=2xy+1+\lambda y=0`

`L_y=x^2+1+\lambda x=0`

`L_\lambda=xy-5=0`


`\underbrace{10}_(2xy)+1+\lambda y=0\implies \lambda=-\frac{11}y`

`xy=5\implies y=\frac5x\implies\lambda=-\frac{11}5x`


`\impliesx^2+1+\left(-\frac{11}5x\right)x=0\implies x^2=\frac56\implies x=\pm√(\frac56)`

`xy=5\implies y=\pm√(30)`

At these points, we get local minima of
`f\left(\pm√(\frac56),\pm√(30)\right)=\pm2√(30)`.

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Another way to do this is to make
`f(x,y)` a function independent of
`y`, which is made possible by the constraint.


`xy=5\implies y=\frac5x`

`\implies f(x,y)=f\left(x,\frac5x\right)=F(x)=6x+\frac5x`

`\implies F'(x)=6-\frac5{x^2}=0\implies x=\pm√(\frac56)`

and so on.