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

Step-by-step explanation:

Consider the function . Note that since the question does not ask to use the lagrange multipliers especificaly, another approach will be used.

NOte that we want to see the behavior of f over the restriction . That is either or .

Let us first replace the equality for x in the function f, then we get the following function

. This is a one variable function, so we can derive find the value of y for which the derivative is 0.

, if we equal it to zero, we get the value of y=0. If we use the second derivative criteria, we have that , which tells us that y=0 is a minimum. Note that if y=0, then we have to critical points, which are (1,0) and (-1,0), both of them are point we f attains it's minimum.

On the other side, let us replace y in the function f on the same fashion. We get the function

.

REcall that

(hence x=0 gives us the critical point)

(hence the critical point is a maximum).

This means that x=0 will give us the maximum for the function f. This occurs over the  points (0,1) and (0,-1).

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