Question

Help! Max and Min?

`z = (7x^2 + 3y^2)e^(-(x^2+y^2))`

Answer 1

`y=7/3x`,
forms tne max / min

Hope the answer is what expected

Explanation:

Given:
`z=(7x^2+3y^2)e^-(x^2+y^2)`

`x=f(x,y)`
Maxima or minima occur when the derivative is zero

`(delz)/dx=(7x^2+3y^2)xx(-e^-(x^2+y^2))(2x)+(e^-(x^2+y^2))xx(7xx2x)`

`=e^-(x^2+y^2)(-2x(7x^2+3y^2)+7xx2x)`

`e^-(x^2+y^2)(-14x^3-6xy^2+14x)`

`e^-(x^2+y^2)ne0`

`-14x^3-6xy^2+14x=0`

`(delz)/dy=(7x^2+3y^2)xx(-e^-(x^2+y^2))(2y)+(e^-(x^2+y^2))xx(3xx2y)`

`=e^-(x^2+y^2)(-2y(7x^2+3y^2)+3xx2y)`

`e^-(x^2+y^2)(-14x^3-6xy^2+6y)`

`e^-(x^2+y^2)ne0`

`-14x^3-6xy^2+6y=0`

The solution of the two conditions

`-14x^3-6xy^2+14x=-14x^3-6xy^2+6y`

`14x=6y`

`x=6/14y=3/7y`

`y=7/3x`