What is the maximum value of this function subject to these conditions

Given that $x, y > 0$ , $x^2+y^2$ is maximum when $x$ and $y$ are maximum. Since $x<3$ and $y<4$ , $\max (x^2+y^2) = (\max (x))^2 + (\max (y))^2 = 2^2+3^2 = 4 + 9 = \boxed{13}$ ,

The feasible set of points given the three conditions are shown in the below figure

The feasible region is OABC

The X, Y points which are in the feasible region which satisfy all specified conditions are

0,0

0,1

0,2

0,3

1,0

1,1

1,2

1,3

2,1

2,2

2,3

It can easily be seen that the maximum value of f(x) occurs at ( x,y) = (2,3) whose value can be calculated as 13.