Find the x coordinate of the absolute maximum for the function f(x)= (2+6lnx) / x?

x>0

xcoordinate of abs max is?

y = 2/x + (6/x) ln x

dy/dx = -2/x^2 +(6/x)(1/x) + ln x (-6/x^2)

zero when
2/x^2 +(6/x^2)lnx = 6/x^2

3 ln x = 3
ln x = 1
x = e

I meant 2+6ln(x) divided by x. I don't know if that's what you were thinking or different?

Yes

[ 2 + 6 ln x ] /x

= 2/x + (6/x) ln x

that makes it easy to take the derivative

Hmm, I tried e but it wasn't right.

zero when

2/x^2 +(6/x^2)lnx = 6/x^2

6 ln x = 4

ln x = 2/3
x = 1.95

To find the x-coordinate of the absolute maximum for the function f(x) = (2 + 6lnx) / x, we need to follow these steps:

1. Start by finding the derivative of the function f(x). The derivative gives us information about the slope (rate of change) of the function at different points.

2. Use the derivative to find the critical points of the function. Critical points are where the derivative is zero or undefined.

3. After finding the critical points, evaluate the function at the critical points and the endpoints of the given domain.

4. Compare the values obtained in step 3 to identify the absolute maximum.

Now let's go through these steps in detail.

Step 1: Finding the derivative
To find the derivative of f(x), we can use the quotient rule. The derivative of f(x) is given by:

f'(x) = [x * (d/dx) (2 + 6lnx) - (2 + 6lnx) * 1] / x^2

Simplifying this expression, we get:

f'(x) = (2/x) - (6lnx/x^2)

Step 2: Finding the critical points
To find the critical points, we set the derivative equal to zero and solve for x:

(2/x) - (6lnx/x^2) = 0

Multiplying through by x^2, we get:

2x - 6lnx = 0

Dividing through by 2, we have:

x - 3lnx = 0

Step 3: Evaluating the function
To identify the critical points and endpoints of the given domain, we need to evaluate the function at these points.

Calculate f(x) at the critical points x and the endpoints of the given domain (x = 0 is not in the domain):

f(x) = (2 + 6lnx) / x

f(x) at x = 0 is undefined.

Now, we need to evaluate f(x) at the critical points x and the endpoints of the given domain.

Step 4: Finding the absolute maximum
Compare the values of f(x) obtained in step 3 to identify the absolute maximum.

The x-coordinate of the absolute maximum is the value of x that gives the highest value of f(x).

By comparing the evaluated values of f(x) at the critical points and endpoints, you can determine the x-coordinate of the absolute maximum. The highest value obtained for f(x) will correspond to the x-coordinate of the absolute maximum.

Note: It is important to consider all points in the domain of the function to find the absolute maximum.

Please note that without actually solving the equation, it is not possible to provide the exact value of the x-coordinate of the absolute maximum for the given function f(x) = (2 + 6lnx) / x.