Find the minimum value of f (x, y) = (x -1)^2 + ( y - 4)^2 + (3- x - 2y)^2.

Find the critical points of f(x, y) :

Subtract the first equation from the second to eliminate x and solve for y :

Solve for x :

So f(x, y) has one critical point at (0, 2).

Compute the Hessian determinant of f(x, y) at this point:

The Hessian has determinant 24 > 0, which indicates a minimum, so the minimum value of f(x, y) is f(0, 2) = 6.