You are responsible for knowing how to solve the following exercises. Please hand in the starred "*" problems.
Please do the following exercises from the text "Foundations of Analysis" by Joseph L. Taylor.
9.4[7*] (This problem is postponed from last week.)
9.5[2, 3*, 4, 6*, 9*, 12]
(The problems from section 9.6 are postponed until next week.)
Please do the following additional exercises.
A*. Find the critical point (s_{0},t_{0}) in the set {(s,t)∈R^{2}:s>0}
for the function with any real A and B>0,
f(s,t)= log(s) + [(t-A)^{2}+B^{2}]/s.
Find the second order Taylor's expansion for f about the point (s_{0},t_{0}). Prove that f has a local minimum at (s_{0},t_{0}).
B*. Let p>1. Find all extrema of the function
f(x)=x_{1}^{2}+...+x_{n}^{2} subject to the constraint |x_{1}|^{p}+...+|x_{n}|^{p}=1.
If 1≤p≤2 show for any x and n that