Maximum and bound for a multivariate function

Quote:

Originally Posted by HTale

Verify that the maximum of

$\displaystyle x(1-x)y(1-y)w(1-w)(1-(1-xy)w)^{-1}$

occurs for $x=y$ , and then that

$\displaystyle \frac{x(1-x)y(1-y)w(1-w)}{(1-(1-xy)w)} \leq (\sqrt 2 - 1)^4$

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 $f(x,y,w) = \frac{x(1-x)y(1-y)w(1-w)}{(1-(1-xy)w)}$ . 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

$(1-2x)(1-(1-xy)w) = (1-x)xyw,\qquad\qquad(1)$
$(1-2y)(1-(1-xy)w) = (1-y)xyw,\qquad\qquad(2)$
$(1-2w)(1-(1-xy)w) = -(1-xy)w(1-w).\qquad(3)$

From (1) and (2) it's easy to see that x=y. From equations (1) and (3) (with y=x) you see that $w = \frac x{1-x}$ . Substitute that into (1) and solve for x, to get $x = \sqrt2-1,\ w=1/\sqrt2$ .

Finally, check that $f(\sqrt2-1,\sqrt2-1,1/\sqrt2) = (\sqrt2-1)^4$ . So that is the maximum value of the function in the cube.