functions What is the monotony of fxxlnxSolutionThe monotony

functions

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

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

To determine whether a function is monotonic, 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 interval (0, 1/e), the function is decreasing, since the first derivative is negative.