Question

Finding local max/min of a multivariable function. I get stuck in the algebra part.

Answer 1

\[F(x,y)= (x-y)(1-xy)\]
FOIL
\[f(x,y) = x-x^2y-y+xy^2\]

Answer 2

\[f_{xx}= 1-2xy+y^2\]
\[f_{yy}= -x^2-1+x\]

Answer 3

So I get stuck when trying to find the critical points. I attempted by solving by elimination but it turned out ugly.

Answer 4

If this is f,\[\rm f(x,y) = x-x^2y-y+xy^2\]then don't you mean to call this f_x? The first derivative?\[\rm f_{x}= 1-2xy+y^2\]

Answer 5

\[\rm f_{xx}=-2y\]

Answer 6

Did you solve \( \nabla f (x,y) = \vec 0 \)

Answer 7

$$ -2xy+y^2+1 = 0
\\ -x^2+2xy-1 = 0
\\ \, \\
\\
x=1, ~ y = 1
\\ x = -1 , ~ y = -1
$$

Answer 8

Oh that's right, I meant first derivatives.

Answer 9

How did you solve for x and y?