How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #f(x) = x ln x#?
In order to find the relative extrema (maxima or minima) for a function, we must find its derivative. This is true because a function is increasing when its derivative is positive and decreasing when its derivative is negative.
However, before we do this, a step students often forget is to consider the domain of the function. Relative extrema cannot exist at points where the function itself does not exist! In this case,
We now find the derivative using the product rule, since
We now consider the "critical points" (points in the domain of
In the domain of
there are no points where
We now look for points in the domain where
We now apply the first derivative test. If we select a number in the domain smaller than
If we select a number in the domain larger than
Since the sign of