Final Exam Solutions and Comments,
Problems 1 and 2
Link to complete solutions below
- Problem 1(b) stated and asked you to prove that
16 ≤ (a2 +
b2 + c2 +
d2)(a-2 +
b-2 + c-2 +
d-2).
Problem 1(c) stated and asked you to prove that
16 ≤ (p + q + r +
s)(p-1 +
q-1 + r-1 +
s-1).
Problem 1(b) is a fairly direct consequence of the Cauchy-Schwartz
inequality, and this was done in class and in the textbook.
THE IMPORTANT POINTS TO NOTE, however, are that
- Problem 1(c) is a direct, simple consequence of
Problem 1(b), and
- you can do Problem 1(c) using Problem 1(b) even if you don't
know how to do 1(b).
In fact, if in Problem 1(c) you think of p as
a2, etc., you immediately get Problem 1(c) from
Problem 1(b). (See the details in the solution below.)
You can just look at 1(c) and see that it is a
consequence of 1(b), provided that in 1(c) the variables are
positive. Problem 1(c) is saying the same thing as Problem
1(b), just with different symbols.
Solutions, Problems 1 and
2 (pdf document).
Go to Solutions, Problems 3-6.
Go to Solutions, Problem 7.
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Last modified Jul 12, 2008 3:32 PM