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No I didn't. I accidently got the answer by guessing. It resulted in an inequality giving me a half correct answer. The equation x^3 = x is the same as x - x^3, but when its in this form then it makes sense with maximums and minimums. x - x^3 will give us the difference and show us how much x exceeds its cube for any value of x. The high point of the graph is the value of x for which it exceeds it the most! :-D
I don't think there is a simpler method. You just sketch the graph to see what it looks like, find the roots algebraically and combine the result(s) with the graph to find the regions. If it is a complicated function, you could use numerical method to estimate the roots to the required degree of accuracy instead of calculating them.
I was solving an equation for velocity, by differentiating the equation of position. I had to find the values of the f(x) for which the velocity is zero, greater then zero, or less then zero. But the velocity equation was a quadratic. So the velocity was zero at two points. But the times when the velocity was greater or less then zero, thats not so simple. The graph of the quadratic will be a parabolla that oppens either upward or downard. It will cross the x axis at the zero's of the equation. Now if it opens upward, then the points between the two zero's will be below the x axis. If it opens downward, the points between the two zero's will be above the x axis.