The derivative of a function has the intermediate value property. Find all values of x where the derivative is zero or where it does not exist. Suppose you find the values where the derivative is zero or it does not exist. Then consider the intervals: . For each interval, if the derivative is positive, it is positive for every value of the interval. If it is negative, it is negative for every value of the interval. So, just pick any point in the interval and plug it into the derivative. That will tell you the sign of the derivative on the entire interval.