Math 113 - Max/Min Worksheet Answers - 10/6
1. (−3, 1).
2. There are two critical points: (1, 3) and (−1, −3). These have fxx = 0 and fxy = ±6. Thus,
D = −36 and so they are both saddle points.
3. (a) There are three critical points with y = 0, they are at x ≈ 1.52569, x ≈ −0.258652 and
x ≈ −1.26704. There are two other critical points, approximately (1.62895, ±1.06252).
(b) The points (1.5269, 0) and (−0.25865, 0) are both saddle points. The remaining three are
local maxes.
(See the attached pages of Maple code that computes these things.)
4. (0, 0) is a local maximum, (−4, 16) is a saddle point and (3, 9) is a saddle point.
2
2
5. One such function is f (x, y) = 31 x3 + xy 2 − x + y. For
we have fx = x + y − 1
this function,
and fy = 2xy + 1. Solving we get the critical points − √12 , √12 and √12 , − √12 .
Both of these points are saddle points, even though D = 0. (I tried to find examples with
D 6= 0, but it turned out to be quite hard. They exist, but I don’t know that you can write
down a polynomial f (x, y) that does it.)
To see they are saddle points, note that
√
1
1
2 2 4 3
+ t
f − √ + t, √ + t =
3
3
2
2
and if t is positive, this number is larger than the value of f , while if t is negative it is smaller.
Similarly,
√
1
1
2 2 4 3
f √ + t, − √ + t = −
+ t
3
3
2
2
1