The monotony of a function is the behavior of the function over specified ranges.

To determine the monotony of a function, we'll have to calculate the first derivative of the function.

f(x) = x*lnx

We'll compute f'(x):

f'(x) = (x*ln x)'

We'll apply the product rule:

f'(x) = (x')*ln x + x*(lnx)'

f'(x) = ln x + x/x

f'(x) = ln x + 1

We recall that the domain of the logarithmic function is (0, +infinite).

We'll determine the critical values for x:

f'(x) = 0

ln x + 1 = 0

ln x = -1

x = e^-1

x = 1/e

For x = 1/e, the first derivative is cancelling.

For x = e => f'(x) = ln e + 1 = 1 + 1 = 2>0

**So, for x>1/e, the function is increasing since f'(x) is positive.**

We'll put x = 1/e^2

f'(x) = ln e^-2 + 1 = -2 + 1 = -1<0

**For x values from the range (0, 1/e), the function is decreasing, since the first derivative is negative.**