Find the extrema of f(x,y)=xy−y−x+1 on the disk x^2+y^2 ≤2?

let g(x,y) = f(x.y) + µ [ x² + y² - 2]

δg/δx = y - 1 + 2 µ x = 0

δg/δy = x - 1 + 2 µ y = 0.....multiply the 1st by y , the 2nd by x and subtract to get

y² - y - x² + x = 0 ---> [ y - 1/2]² - [ x - 1/2 ]² = 0------> y - 1/2 = ± [ x - 1/2]...

.if + then x = y ; if - then y = - x + 1...

thus (1,1) , (0.5 + 0.5 √3, 0.5 - 0.5 √3) , ( 0.5 - 0.5 √3 , 0.5 + 0.5 √3 )..all on the boundary

inside the circle you have f_x = y - 1 ; f_y = x - 1----> x = y = 1 { which is not inside the circle }