What is the monotony of f(x)=x*lnx?

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.