Find the absolute extrema of the function on the interval [2, 9]. (Round your answers to the nearest hundredth.)

g(x)=x/(ln(x))
Absolute minimum: at x =
Absolute maximum: at x =

To find the absolute extrema of the function g(x) = x/ln(x) on the interval [2, 9], we need to first find the critical points and the endpoints of the interval.

1. Critical Points:
A critical point is a point where the derivative of the function is either zero or does not exist. To find the critical points of g(x), we need to find where the derivative is equal to zero or undefined.

First, let's find the derivative of g(x):

g'(x) = (d/dx)(x/ln(x))

Using the quotient rule, we have:

g'(x) = [(ln(x))(1) - (x)(1/x)] / (ln(x))^2
= [ln(x) - 1] / (ln(x))^2

To find the critical points, we need to solve the equation g'(x) = 0:

ln(x) - 1 = 0

Solving for x:

ln(x) = 1

Taking the exponential of both sides:

x = e

Therefore, x= e is a critical point.

2. Endpoints of the Interval:
The given interval is [2, 9]. We need to evaluate the function g(x) at the endpoints of the interval.

g(2) = 2/ln(2)
g(9) = 9/ln(9)

3. Absolute Extrema:
To find the absolute maximum and minimum of g(x) on the interval [2, 9], we need to compare the values of g(x) at the critical points and the endpoints.

Evaluate g(x) at the critical point:
g(e) = e/ln(e) = e

Evaluate g(x) at the endpoints:
g(2) = 2/ln(2)
g(9) = 9/ln(9)

Now we compare these values:

The smallest value among g(2), g(9), and g(e) is the absolute minimum, and the largest value is the absolute maximum.

To find the absolute minimum, compare g(2), g(9), and g(e), and choose the smallest value.

To find the absolute maximum, compare g(2), g(9), and g(e), and choose the largest value.

Finally, round the answers to the nearest hundredth.