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

Without using Lagrange multipliers: Consider the same problem in polar coordinates, where we take

to restrict to the circle , or . Then

which is a function of alone; denote this function by .

Find the critical points:

where is any integer. In one revolution of the circle, we get 4 critical points for , which correspond to the points (1, 0), (0, 1), (-1, 0), and (0, 1) in Cartesian coordinates and to the extreme value of 3.

Take the second derivative:

Then and are both positive, while and are both negative, which indicates (1, 0) and (-1, 0) are minima, and (0, 1) and (0, 1) are maxima.

f(x) is the value of the function. b is the value of the function when x equals zero or the y-coordinate of the point where the line crosses the y-axis in the coordinate plane.

step-by-step explanation:

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