Find all the stationary points of f and determine whether each is a local maximum, a
local minimum or a saddle point.
8. Let f(x,y) = . - y, g(x,y) = x2 + y2 - 1. This question is about finding the maximum
and minimum value of f(x,y) subject to the constraint g(x,y)
(a) Draw some level curves of f (x,y) and in the same picture draw the constraint set
g (x, y) = 0. Mark on your picture the points at which you think f (x,y) will assume
its constrained maximum and minimum values, and explain how you estimated the
position of these points.
(b) Use the Lagrange multiplier condition to find the exact coordinates of the points you
located in part (a), and hence find the constrained maximum and minimum values
9. Find the maximum and minimum values of
f(x,y) = x2 + x + 2y2
on the unit circle x2 + y2 = 1.
10. Find the minimum and maximum values of
f x, y) = x2 - xy + y2
inside the quarter circle given by r²+y²1, x, y > 0.
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