If x and y are real numbers such that `16^(x^(2)+y)+16^(x+y^2)=1`, then find the values of x and y.

Explanation

Using A.M `ge` G.M we get
`(16^(x^(2) + y) + 16^(x + y)^(2))/(2) ge sqrt(16^(x^(2) + y + x + y^(2))`
`:. (1)/(2) ge 2^(2(x^(2) + y + x + y^(2)))`
`implies 2^(2(x^(2) + y + x + y^(2))) le 2^(-1)`
`implies x^(2) + x + y^(2) + y le (-1)/(2)`
`implies x^(2) + x + (1)/(4) + y^(2) + y + (1)/(4) le 0`
`implies (x + (1)/(2))^(2) + (y + (1)/(2))^(2) le 0`
`implies (x + (1)/(2))^(2) + (y + (1)/(2))^(2) = 0`
`implies x = - (1)/(2), y = - (1)/(2)`