Determine the coordinates of the extreme point of y = xlnx .

To determine the coordinates of the extreme point of the
given function, we'll differentiate the function.

f(x) = y
= xlnx

Since the expression of the function is a product,
we'll apply the product rule:

(u*v)' = u'*v +
u*v'

We'll note u = x => u' =
1

We'll note v = ln x => v' =
1/x

u'*v + u*v' = 1*lnx +
x*(1/x)

We'll eliminate like
terms:

u'*v + u*v' = ln x +
1

To calculate the coordinates of the extreme point, we'll
have to calculate the values of x for the derivative of the function is
cancelling;

ln x + 1 = 0

ln x
= -1

x = e^-1

x =
1/e

We'll calculate f(x) for x =
1/e

f(1/e) = (1/e)*ln
e^-1

f(1/e) = -1*(1/e)*ln
e

f(1/e) =
-1/e

The coordinates of the extreme point
are: (1/e , -1/e).