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?

1 Answer
Jan 26, 2017

f(x) is decreasing on (0,1/e) and increasing on (1/e, oo). Therefore, this point is a relative minimum. There are no relative maxima.

Explanation:

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, lnx does not exist when x<=0. Therefore, the domain of f(x) is (0,oo).

We now find the derivative using the product rule, since f(x) is a product of two functions:

f(x)=xlnx
f'(x)=x*(1/x) +lnx*1
f'(x)=1+lnx

We now consider the "critical points" (points in the domain of f(x) where f'(x)=0 or does not exist). These would be places where f' could change signs, causing f to change directions.

In the domain of (0,oo):
there are no points where f' does not exist.
We now look for points in the domain where f'(x)=0.
f'(x)=0
1+lnx=0
lnx=-1
x=e^(-1)=1/e=0
We now apply the first derivative test. If we select a number in the domain smaller than x1/e=0.36787944 . . . such as x=0.1,
f'(0.1) =1+ln(0.1)=-1.30258 . . ..
If we select a number in the domain larger than x=1/e such as x=5,
f'(5) =1+ln(5)=2.68943 . . ..

Since the sign of f'(x) changes from negative to positive at x=1/e, the direction of f(x) changes from decreasing to increasing at this point, making it a relative minimum by the first derivative test. In fact, since it is the only critical number, it is also the absolute minimum.