If `(log)_(10)(x^3+y^3)-(log)_(10)(x^2+y^2-x y)lt=2, and x ,y` are positive real number, then find the maximum value of `x ydot`

Answer

600

Explanation

`log _10(x^3+y^3)-log_10(x^2+y^2-xy)le 2`
`therefore log_10(x+y)(x^2+y^2-xy)-log _10(x^2+y^2-xy)le 2`
`therefore log)10(x+y)le 2`
`therefore x+yle 100`
Now `(x+y)/(2) ge(xy)^(1//2)`
or `xyle ((x+y)/(2))^2le ((100)/(2))^2`
`therefore xyle 2500`
`therefore xy_("max")=2500`.