M3220-1 Ninth Homework Assignment

MATH 3220 § 1 NINTH HOMEWORK ASSIGNMENT Due Friday,

A. Treibergs November 2, 2007.

MATH 3220 § 1	NINTH HOMEWORK ASSIGNMENT	Due Friday,
A. Treibergs		November 2, 2007.

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 (s₀,t₀) in the set {(s,t)∈R²:s>0} for the function with any real A and B>0,
f(s,t)= log(s) + [(t-A)²+B²]/s.
Find the second order Taylor's expansion for f about the point (s₀,t₀). Prove that f has a local minimum at (s₀,t₀).

B*. Let p>1. Find all extrema of the function f(x)=x₁²+...+x_n² subject to the constraint |x₁|^p+...+|x_n|^p=1. If 1≤p≤2 show for any x and n that
n^(p-2)/(2p) (|x₁|^p+...+|x_n|^p)^1/p ≤ (x₁²+...+x_n²)^1/2 ≤ (|x₁|^p+...+|x_n|^p)^1/p.

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 (s₀,t₀)* in the set {(s,t)∈R²:s>0} for the function with any real A and B>0, f(s,t)= log(s) + [(t-A)²+B²]/s. Find the second order Taylor's expansion for f about the point (s₀,t₀). Prove that f has a local minimum at (s₀,t₀). B. Let p>1. Find all extrema of the function f(x)=x₁²+...+x_n²* subject to the constraint \|x₁\|^p+...+\|x_n\|^p=1. If 1≤p≤2 show for any x and n that n^(p-2)/(2p) (\|x₁\|^p+...+\|x_n\|^p)^1/p ≤ (x₁²+...+x_n²)^1/2 ≤ (\|x₁\|^p+...+\|x_n\|^p)^1/p.