@SteveRoth if we are trying to maximize y=-(x^2)+x, though, we find an inverted U shape. initially as x rises, the x term overwhems the -(x^2) term. but at a certain point (at x=1/2) the negative term matches then overwhelms the positive. the solution is interior: neither going to zero (if that's a lower constraint) nor rising indefinitely towards infinity maximizes the function. there's a value in the middle that balances the terms, or (in a perhaps risky intuition) the tradeoffs. 3/