EXAMPLE 2 Find the extreme values of the function f(x, y) = 3x2 + 4y2 on the circle x2 + y2 = 1. SOLUTION We are asked for extreme values of f subject to the constraint g(x, y) = x2 + y2 = 1. Using Lagrange multipliers, we solve the equations ∇f = λ∇g and g(x, y) = 1, which can be written as fx = λgx fy = λgy g(x, y) = 1 or as (1) Correct: Your answer is correct. = 2xλ (2) Correct: Your answer is correct. = 2yλ (3) x2 + y2 = 1. From (1) we have x = 2 Incorrect: Your answer is incorrect. or λ = 3. If x = , then (3) gives y = ±1. If λ = 3, then y = from (2), so then (3) gives x = ±1. Therefore f has possible extreme values at the points (0, 1), (0, −1), (1, 0), and (−1, 0). Evaluating f at these four points, we find that