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sniffnoyThis was a problem on the analysis midterm today:
a. Prove or disprove: p:R→R a polynomial, p bounded below. Then p takes a minimum.
b. Prove or disprove: p:R²→R a polynomial, p bounded below. Then p takes a minimum.
The first, of course, is true, but interestingly enough the second isn't. Mr. Sally gave us the counterexample after the test was over: (xy-1)²+x². It's obviously ≥0, and on xy=1, it's x², so it gets arbitrarily close to 0, but if x=0 then it's 1, so it's never 0.
a. Prove or disprove: p:R→R a polynomial, p bounded below. Then p takes a minimum.
b. Prove or disprove: p:R²→R a polynomial, p bounded below. Then p takes a minimum.
The first, of course, is true, but interestingly enough the second isn't. Mr. Sally gave us the counterexample after the test was over: (xy-1)²+x². It's obviously ≥0, and on xy=1, it's x², so it gets arbitrarily close to 0, but if x=0 then it's 1, so it's never 0.
no subject
Date: 2006-05-05 06:40 pm (UTC)no subject
Date: 2006-05-07 09:39 am (UTC)