Curl function - help!

I'm trying to find a function $A(r)$ that fits the following conditions:

$A(0) = 0$
$A(0<r<1) > 0$
$A(1) = 0$

And if $B(r) = \frac{1}{r}\left(\frac{\partial}{\partial r} r A(r)\right)$:

$B(0) = 1$
$B(1) = 0$
$B(0<r<1) > 0$

(i.e. A(r) has to be zero at r = 0 and r = 1, and B(r) has to have a maximum at r = 0 and has to be 0 at r = 1, and both functions must be greater than or equal to zero for r between 0 and 1).

Is this possible? So far I've been using a function based on $\cos^2(\frac{\pi r}{2 r_0})$ as $B(r)$ which satisfies all of the above apart from $A(1)=0$ but I really need to find something that fulfils this condition too.

Here's a hint: take your differential expression for B in terms of A and invert it by integration to get A in terms of B. Then note that A(1) is an integral of stuff that is strictly poisitive and therefore cannot be zero.