I think you need to find an upper bound for K, not a lower bound.
But you can't get a bound of any kind by setting x to the same value when you have the product of a decreasing and an increasing function. For example, if you have y=x(1-x) on [0,1], if you set x=0 across the board, you get a bound of zero, but y goes above zero. The correct procedure in this simplistic case is to recognize that x is always less than or equal to 1, and 1-x is always less than or equal to 1. So x(1-x) is always less than or equal to 1 times 1, which is 1. In this case we know the function has a maximum of , so it is indeed always less than or equal to 1.