1. The problem statement, all variables and given/known data Show that , the maximum value of function f(x,y) = x^2 + y^2 is 70 and minimum value is 20 in constraint below. 2. Relevant equations Constraint : 3x^2 + 4xy + 6y^2 = 140 3. The attempt at a solution Book's solution simply states the Lagrange rule as : h(x , y ; L) = x^2 + y^2 - L(3x^2 + 4xy + 6y^2 - 140) Takes partial derivatives for x , y and L. h's partial derivative for x = 2x + L(6x + 4y) = 0 h's partial derivative for y = 2y + L(4x + 12y) = 0 h's partial derivative for L = 3x^2 + 4xy + 6y^2 -140 = 0 Then he takes the coefficients determinant of first two equations ( h derv for x , h derv for y) |1+3L 2L | |2L 1+6L | And makes this determinant equal to zero , finds values for L. The part i don't understand is , why he uses Cramer rule to the first two equations and equals it to zero ?