Question

(1 point) Consider the function defined by

?(?,?)=??(9?2+5?2)?2+?2F(x,y)=xy(9x2+5y2)x2+y2

except at (?,?)=(0,0)(x,y)=(0,0) where ?(0,0)=0F(0,0)=0.
Then we have
∂∂?∂?∂?(0,0)=∂∂y∂F∂x(0,0)=
∂∂?∂?∂?(0,0)=∂∂x∂F∂y(0,0)=
Note that the answers are different. The existence and continuity of all second partials in a region around a point guarantees the equality of the two mixed second derivatives at the point. In the above case, continuity fails at (0,0)(0,0).

(1 point) Consider the function defined by F(x, y) = xy(9x2 + 5y2) x2 + y2 except at (x, y) = (0,0) where F(0,0) = 0. Then we

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Answer #1

Given 2 U ONE OF (ory [ylaxtasy) + xy (18x)] dne mey (9x 5y) (22) 2 (ox?ry 2) JF dy Gy![*1*zydomy] - xy (9x², by ?) (ay) (s

Similarly of dre be Simplified to Jf 1 (984 + 220 ди 2 2200²4² + 5y4) ( ) of) (och ry, [(an? 4 22n?y? By ) + y (Aarly + » 20

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(1 point) Consider the function defined by ?(?,?)=??(9?2+5?2)?2+?2F(x,y)=xy(9x2+5y2)x2+y2 except at (?,?)=(0,0)(x,y)=(0,0) where ?(0,0)=0F(0,0)=0. Then we have...
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