Find the monotony intervals of a function f = 18x^2 - lnx .

To determine the inervals of monotony of a function, we
have to calculate it's derivative and to determine the intervals where the derivative is
positive or negative.

If derivative is positive over an
interval, then the function is increasing over that
interval.

If derivative is negative over an interval, then
the function is decreasing over that interval.

Now, we'll
calculate the derivative of the function:

f'(x) = (18x^2 -
ln x)'

f'(x) = 18*2*x -
1/x

f'(x) = 36x - 1/x

We'll
calculate the LCD:

f'(x) =
(36x^2-1)/x

We'll verify where the numerator is positive or
negative, if x>0 (the constraint of the existance
of logarithm).

36x^2-1 = 0

It
is a difference of squares:

(6x-1)(6x+1) =
0

We'll put each factor from the product as being
0.

6x-1 = 0

We'll add 1 both
sides:

6x = 1

We'll divide by
6;

x = 1/6

6x+1 =
0

6x = -1

x =
-1/6

Now, we'll discuss the values of f'(x), but only for x
values bigger than 1/6, because -1/6 does not belong to the admissible interval
(0,+inf.).

f'(x) is positive over the interval [1/6, +inf.)
and is negative over the interval (0,1/6).

The function is
increasing over [1/6, +inf.) and is decreasing (0,1/6).