Friday, 27 March 2015

calculus - Prove an inequality involving positive real numbers Lagrange multipliers




This is from Apostol's Calculus, Vol. II, Section 9.15 #11:




Find the maximum of f(x,y,z)=logx+logy+3logz on that portion of the sphere x2+y2+z2=5r2 where x>0,y>0,z>0. Use the result to prove that for real positive numbers a,b,c we have
abc327(a+b+c5)5




I had no trouble with the first part, using Lagrange's Multipliers. The maximum of f subject to this constraint is f(r,r,3r)=5logr+3log3, and this answer matches the book's.




Now I see how we can take f(a,b,c)=log(abc3). Then define r>0 by a2+b2+c2=5r2r=a2+b2+c25, so we can conclude that



abc333/2(a2+b2+c25)5/2



But this is a looser bound (for some numbers) than the one suggested by the text. In particular, if we consider a=14,b=1,c=1, then 27(a+b+c5)5<33/2(a2+b2+c25)5/2
so I have the feeling that "I can't get there from here", at least not using the method suggested. Am I correct?



Hints > Full Answers


Answer



I think I'm going to delete this question - I realize no one (except maybe Apostol, and perhaps not even him) can really answer this question. Perhaps what was intended was that I should realize that a similar approach would yield the inequality desired, and in fact it does:




Maximize f(x,y,z)=logx+logy+3logz subject to the constraint that x+y+z=5r, and then use this result to deduce that, for real numbers a,b,c we have that abc327(a+b+c5)5



This actually works in a straightforward way. It's anyone's guess, I suppose, whether the author's "use this result" was meant in the broadest sense (i.e. generalize a strategy from the first result), or whether it was a typo in either the constraint or the inequality.



If no one has any complaints, I will delete this question tomorrow morning.


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