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.

answer: there are x+184 pages in olivia's book.

step-by-step explanation:

let the total number of pages in the book which olive has already read be x.

then if olive has 184 pages left to read, then the total number of pages are in olive's book= pages she has read+pages she has left to read

therefore, the total number of pages in olivia's book=

hence, there are x+184 pages in olivia's book.