find max and min of f(x,y) = (x - y)^2 xy
- print Print
- list Cite
Expert Answers
calendarEducator since 2011
write5,349 answers
starTop subjects are Math, Science, and Business
You need to expand the square using the formula `(a-b)^2 = a^2 - 2ab + b^2` such that:
`f(x,y)= (x^2 - 2xy + y^2)xy => f(x,y) = x^3y - 2x^2y^2 + xy^3`
You need to evaluate the critical points, hence, you need to solve the equations `f_x=0` and `f_y=0` such that:
`f_x = 3x^2y - 4y^2x + y^3 => 3x^2y - 4y^2x + y^3 = 0`
`y(3x^2 - 4xy + y^2) = 0 => y = 0`
`3x^2 - 4xy + y^2 = 0 => 4xy = 3x^2 + y^2`
`f_y = x^3 - 4x^2y + 3xy^2 => x^3 - 4x^2y + 3xy^2 = 0`
`x(x^2 - 4xy + 3y^2) = 0 => x = 0`
Substituting `3x^2 + y^2` for `4xy` yields:
`x^2 -(3x^2 + y^2) + 3y^2 = 0`
`x^2 - 3x^2 + 3y^2 - y^2 = 0`
`-2x^2 + 2y^2 = 0 => -2(x^2 - y^2) = 0`
Converting the difference of squares into a product yields:
`x^2 - y^2 = (x-y)(x+y)`
`x^2 - y^2 = 0 => (x-y)(x+y) = 0 => x = +-y`
Hence, evaluating the critical points of the given function yields that the only critical point is (0,0).
Related Questions
- find the values of x where the function: y= x^3 -3x^2 + 2 reaches a max and min, also calculate...
- 1 Educator Answer
- Find both first partial derivatives.?f(x,y)=xy/(x^2+y^2) f x(x,y)= f y(x,y)=
- 1 Educator Answer
- `x^2 + xy - y^2 = 4` Find `(dy/dx)` by implicit differentiation.
- 1 Educator Answer
- Find the slope of the curve x^2 + xy + y^2 = 7 at (1,2)
- 1 Educator Answer
- Find the inverse function of f(x)=x^3+2?
- 1 Educator Answer
Unlock This Answer Now
Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.