This is the function: $f(x,y)=xye ^{-x-y}$ I have trouble finding critical points and fxx, fyy and fxy. I think fx is $fx=(x-1)y(-e ^{-x-y)}$ $fy=x(y-1)(-e ^{-x-y)}$
finding and analyzing critical points
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calculus
2 Answers
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The partials are ${f_x} = y{e^{ - x - y}}(1 - x)$ ${f_y} = x{e^{ - x - y}}(1 - y)$ so your answers are correct. With these you can find ${f_{xx}}$, ${f_{yy}}$ and ${f_{xy}}$ using the product rule and the chain rule.
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0Yes. Take $y{e^{ - x - y}}$ and $(1 - x)$, then use the product rule and the chain rule. Then take $x{e^{ - x - y}}$ and $(1 - y)$ to compute ${f_{yy}}$. Do not forget to factor. – 2012-10-22
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Notice the exponential function is always positive(or negative). So, for $f_x = 0$, you have either $x = 0$ or $y =0$ and in the $f_y = 0$, you have $x = 0$ or $y=1$. Now, what are the critical points???? You know should be able to find them.
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0(0,0), (1,1)? but i have a hard time finding fxx and fyy and fxy now. – 2012-10-22