The vertex of a parabola is the extreme point of the function. To determine the extreme point, we'll determine the critical points that are the roots of the first derivative of the function.

f(x)=3(2x-1)^2+(x+1)^2

We'll calculate the first derivative of f(x), with respect to x.

f'(x) = 6(2x - 1)*(2x-1)' + 2(x+1)*(x+1)'

f'(x) = 12(2x - 1) + 2(x+1)

We'll remove the brackets:

f'(x) = 24x - 12 + 2x + 2

We'll combine like terms:

f'(x) = 26x - 10

Now, we'll put f'(x) = 0, to determine the critical point:

26x - 10 = 0

We'll simplify by 2:

13x - 5 = 0

We'll add 5 both sides and we'll divide by 13:

x = 5/13

The critical point of f(x) is x = 5/13. The extreme point of f(x), namely the vertex of the parabola, is f(5/13).

f(5/13) = 3(10/13 - 1)^2+(5/13 + 1)^2

f(5/13) = 27/169 + 324/169

f(5/13) = 351/169

**The coordinates of the vertex are: V(5/13 , 351/169).**