Mathematics I (Calculus) - Old Questions

15.  Find the extreme values of the function f(x, y) = x² + 2y² on the circle x² + y² = 1. 

Given,

And, let

By method of Lagrange’s multiplier, for some scalar 

This implies,

This gives

From (ii) we have x = 0 or λ = 1. If x = 0, then (i) gives y = ±1. If λ = 1, then y = 0 from (iii), so then (i) 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

f(0, 1) = 2

f(0, −1) = 2

f(1, 0) = 1

f(−1, 0) = 1

Therefore, the maximum value of f on the circle x ² + y ² = 1 is f(0, ±1) = 2 and the minimum value is f(±1, 0) = 1.