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anonymous
 4 years ago
locate and classify all critical points for f(x,y)=4ypower2 x  2yxpower2 + 3xy
anonymous
 4 years ago
locate and classify all critical points for f(x,y)=4ypower2 x  2yxpower2 + 3xy

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Mr.Math
 4 years ago
Best ResponseYou've already chosen the best response.0We want to find the critical points of \(f(x,y)=4y^2x2yx^2+3xy\). The critical points of such a polynomial is the points satisfying \(f_x=0,f_y=0\) simultaneously. So lets first find the partial derivatives: \(f_x=4y^24xy+3y\) and \(f_y=8xy2x^2+3x\). \[f_x=0 \implies 4y^24xy+3y=0 \implies y(4y4x+3)=0\] \(\large \implies y=0 \text{ or } y=x\frac{3}{4}.\)

Mr.Math
 4 years ago
Best ResponseYou've already chosen the best response.0Substituting \(y=0\) into \(f_y=0\) gives us: \[f_y_{y=0}=2x^2+3x=0 \implies x(2x3)=0 \implies x=0 \text{ or } x=\frac{3}{2}.\] Substituting \(y=x\frac{3}{4}\) gives: \[\small 8x^26x2x^2+3x=6x^23x=3x(2x1)=0 \implies x=0 \text{ or } x=\frac{1}{2}.\] \(x=0 \implies y=\frac{3}{4} \text{ and } x=\frac{1}{2} \implies y=\frac{1}{4}\). Thus the critical points are \((0,0), (\frac{3}{2},0), (0,\frac{3}{4}), (\frac{1}{2},\frac{1}{4}). \)
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