If y=a/x + x/b has a local minimum at (4,4), state the value of a+b

You know two things. First, when X=4, then Y=4 so:

4 = a/4 + 4/b

Also to have a local minimum, the derivative is zero when X=4.

y' = -a/x^2 + 1/b
0 = -a/16 + 1/b

You now have two equations in two variables and can solve.