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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.
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