Determine the point where function output will go from positive to negative I have a function that is like:
f(x) = c - x^2 (c = some constant positive integer, x = +ve integer >= 0)

The output of this function, goes from positive to negative as x -> +infinity. 


*

*Is there a way to directly figure out the x which produces last of the positive outputs before entering in the negative domain? 

*Also, if I use an iterative method to locate such a point, how should I find the right increment to go with so that I can reach that value of x as quick as possible? Right now, I am initializing x = floor(sqrt(c) * 0.9) and using dumb +1 increments to x to reach the point where f(x) enters the negative range.

*An observation: f(x) seems to behave in a weird way for c > 10e40 (i.e. when we initiate x = floor(sqrt(c) * 0.9) it gives out a value way too far from the desired point) and you can imagine how long the +1 increment takes to get to the desired output with such large values of c.. ;-(.
Please help. thanks.

If we solve the quadratic equation $c-x^2=0$ for a real number $x$, we get $x=\sqrt c$ or $x=-\sqrt c$. Since we are looking at the branch $x\ge 0$, the required solution is the largest integer not exceeding $\sqrt c$. For example, if $c=2$ then $\sqrt c=1.414\ldots$ and the solution would be $1$.
