I agree with you in that if X and Y are independent variables then the cross partial derivative should have a Y^(b-1) term instead of a Y^(b-2) term.
If L= X^a Y^b + lamda (p1(m - X) + p2(n - Y))
dL/dx1= aX^a-1 Y^b - lamda p1
dL/dy1= bX^a Y^b-1 - Lamda p2
The cross partial derivative d^2L/ Dx1Dy1 = ab(x^a-1 y^b-2)
Where does the y^b-2 come from is the cross derivative? why is it not just y^b-1
Any explanation would be a big help. Cheers