The Idea here is we want to find the maximum value of the derivative on the square because it will give us an upper bound on how fast the function can grow!

Here is an example lets us use the same region R and initial condition, but look at this problem

we identify that

So on R the maximum will occur at the points these give

so the maximum slope of this function is since we were given the intial point we can bound this by a linear function

So the biggest interval we know for sure that the solution stays between