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How to do the high number 12 problem?
The answer is B.

analyse

(1) continuity is obvious.

Elementary function, (0,0) is in the defined area.

(2) The existence of partial derivatives.

lim(△x→0)[f(0+△x,0)-f(0,0)]/△x

=lim(△x→0)(0-0)/△x

=0

∴ partial derivative FX (0 0,0) = 0

Similarly, the partial derivative fy (0 0,0) = 0.

(3) Non-differentiable.

△z=f(△x,△y)-f(0,0)=√|△x △y|

fx(0,0)△x+fy(0,0)△y=0

△z-[fx(0,0)△x+fy(0,0)△y]=√|△x △y|

This is not ρ=√(△x? +△y? ) is infinitely small in high order,

Therefore, f(x, y) is nondifferentiable at (0,0).