Given:
Linear density of the rod, λ = 2x
Length of the rod, L = 3m

To find:
The position of the center of mass

Solution:

1. Determining the mass element:
The linear density of the rod is given by λ = dm/dx, where dm is the mass element and dx is the length element.
Given that λ = 2x, we can write dm/dx = 2x.

2. Integrating the mass element:
To find the total mass of the rod, we integrate the mass element over the length of the rod.
∫dm = ∫2x dx
m = ∫2x dx
m = x^2 + C
where C is the constant of integration.

3. Determining the value of the constant of integration:
To find the value of the constant of integration, we use the given length of the rod.
When x = 0, m = 0 (since the mass of the rod is zero at the starting point).
Therefore, C = 0.

4. Expressing the mass element in terms of x:
Using the value of C, we have m = x^2.

5. Determining the center of mass:
The center of mass of the rod is given by the equation:
xcm = (∫x dm) / (∫dm)
where xcm is the position of the center of mass.

Substituting the expression for dm and m from steps 1 and 4 respectively, we have:
xcm = (∫x (x^2) dx) / (∫(x^2) dx)

Simplifying the integrals, we have:
xcm = (∫(x^3) dx) / (∫(x^2) dx)
xcm = (x^4)/4 / (x^3)/3
xcm = 3x/4

6. Substituting the length of the rod:
The length of the rod is given as L = 3m.
Substituting L into the equation for xcm, we have:
3 = 3x/4
x = 4/3

Answer:
The position of the center of mass is x = 4/3m.

Note:
None of the given options (A, B, C, D) matches the calculated position of the center of mass (x = 4/3m).