I set up a sample problem similar to your description, and it seems to work
fine (I use Excel 2000).

B1: value
B2: =-1*((B1-5)^2)+10

This is a parabola, with the maximum at 5 and cupped downward, like you
mentioned.

Solver Parameters:
---------------------
Set Target Cell: $B$2
By Changing Cells: $B$1
Subject to the Constraints:
$B$1 <= 10
$B$1 = integer
$B$1 = 0

Options: (I left everything at the defaults)
--------------------------------------------
Max Time: 100 seconds
Iterations: 100
Precision: 0.000001
Tolerance: 5%
Convergence: 0.0001

All checkboxes OFF (Assume Linear Model, Assume Non-Negative, Use Automatic
Scaling, Show Iteration Results).

Estimates: Tangent
Derivatives: Forward
Search: Newton

I could start with cell $B$1 at 0 or 10, and it would converge to
4.9999999824217 in about 1 second.

--
Regards,
Bill Renaud