Minimizing Diagonals of rectangles

Of all rectangles with given area, , which has the shortest diagonals?

I'm not sure what this question is asking, but I tried. I'll show my work, Please tell me if I did it correctly or help me if I didn't.

x = length

y = width

A = area

The diagonal would be .

Because I want to minimize the diagonals, I solved for y in the area function and substituted it into the formula for the diagonal.

I'll take the derivative of the diagonal and set it equal to zero to minimize it (here i treated A as a constant... I'm not sure if i can do this.. this may be my mistake.. I'm not sure).

Plug it back in to the area to find the other dimension.

The dimensions should be .

Is this what they wanted? And is this correct? I have no clue =\