This is modeled as a LaGrange Multiplier problem with f(x,y) = 5x + y, g(x,y) = x2 + 36y2 −1 and the aim is to seek the values of x, y, and λ that satisfy the equations ∇f = λ∇g: 5i + j = 2xλi + 72yλj; g(x,y) = 0.
The gradient equation implies that λ is not 0 and gives x = 5/2λ & y = 1/72λ. These equations indicate that x and y have the same sign. Place these expressions for x and y into g(x,y) = 0 to obtain
(5/2λ)2 + 36(1/72λ)2 − 1 = 0.
25/4λ2 + 36/722λ2 = 1 will give λ as ±√901/12. This will determine x as ±30√901/901 and y as ±√901/5406.
It follows that f(x,y) = 5x + y has extreme values at the points (x,y) = ±(30√901/901,√901/5406). There are only two points instead of four points because x and y have the same sign. By calculating the value of
5x + y at the points ±(30√901/901,√901/5406), it is seen that its maximum and minimum points on the ellipse x2 + 36y2 = 1 are 5(30√901/901) + √901/5406 which gives √901/6 (maximum) and
5(-30√901/901) + -√901/5406 equal to -√901/6 (minimum).
