Math 164, Lecture 2, Vese
Homework #8, due on Friday, March 10, 2006
Problems:
[1] Determine if
f(x1, x2) = 2x2
1−3x1x2+ 5x2
2−2x1+ 6x2
is convex, concave, both, or neither for x∈R2.
[2] Find the first 3 terms of the Taylor series for
f(x1, x2) = 3x4
1−2x3
1x2−4x2
1x2
2+ 5x1x3
2+ 2x4
2
at the point x0= (1,−1)T. Evaluate the series for p= (.1, .01)Tand compare with the value
of f(x0+p).
[3] Consider the following function
f(x) = 15 −12x−25x2+ 2x3.
(a) Use the first and second order derivatives to find the local maxima and local minima
of f.
(b) Show that fhas neither a global maximum nor a global minimum.
[4] Consider the function
f(x) = 8x2
1+ 3x1x2+ 7x2
2−25x1+ 31x2−29.
Find all stationary points of this function, and determine whether they are local mini-
mizers and maximizers. Does this function have a global minimizer or a global maximizer
?
[5] Use Newton’s method to solve
minimize
f(x1, x2) = 5x4
1+ 6x4
2−6x2
1+ 2x1x2+ 5x2
2+ 15x1−7x2+ 13
Use the initial guess (1,1)T. Make sure that you have found a minimum and not a maximum.
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