There is an important condition missing from this question. As it stands, the function is unbounded. For example, it goes to +∞ if x=y=½ and w→4/3 from above. I think that the missing condition must be that x, y and w must all lie in the unit interval [0,1].

Let . Then f is identically zero on the boundary of the unit cube, so its maximum must occur at a critical point inside the cube. So put the three partial derivatives equal to 0. Leaving out some details, this gives the equations

From (1) and (2) it's easy to see that x=y. From equations (1) and (3) (with y=x) you see that . Substitute that into (1) and solve for x, to get .

Finally, check that . So that is the maximum value of the function in the cube.