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## Homework Statement

Show that [tex]\frac{2x+1}{[(x+1)^2+y^2]^{3/2}}+\frac{x-1}{[(x-1)^2+y^2]^{3/2}} < 0 [/tex] for [tex] 0 < x < 1 [/tex] and [tex] 0 < y < \frac{x}{\sqrt{3}}+\frac{1}{\sqrt{3}} [/tex].

## Homework Equations

## The Attempt at a Solution

I've confirmed by graphing in Maple.

It's easy to see that [tex] (x+1)^2+y^2 > (x-1)^2+y^2 [/tex], and therefore [tex]\frac{2x+1}{[(x+1)^2+y^2]^{3/2}}<\frac{2x+1}{[(x-1)^2+y^2]^{3/2}}[/tex]. Unfortunately, this is a dead end because when you add the two fractions, the numerator becomes 3x, which is strictly positive.

I've tried differentiating both terms with respect to x to show that the negative term is decreasing faster than the positive term; since at x=0 the sum is 0, this would imply that the sum is always negative for x>0, but I've had no luck.

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