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Math Help - Langrangean: Simple proof question, answer completely eluding me

  1. #1
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    Langrangean: Simple proof question, answer completely eluding me

    Hi everyone,

    I was hoping someone could help me with the second part of this question. I have found the minumum using the lagrangean solution and found x=y=x=c/3 however with the proof don't both the left hand side and right hand side equal eachother exactly!?
    Many thanks for any impending help!
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  2. #2
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    Re: Langrangean: Simple proof question, answer completely eluding me

    If x= y= z= c/3, then the two sides are equal but that is not true for other values of x, y, z. For example, if x= y= c/4 and z= c/2, then 1/x+ 1/y+ 1/z= 4/c+ 4/c+ 2/c= 10/c and so ((1/3)(1/x+ 1/y+ 1/z))^{-1}= 3c/10< c/3. The point is that since x= y= z= c/3 minimizes 1/x+ 1/y+ 1/z, it maximizes (1/x+ 1/y+ 1/z)^{-1} and so maximizes 3(1/x+ 1/y+ 1/z)^{-1}= ((1/3)(1/x+ 1/y+ 1/z))^{-1/3}. Since that expression is equal to (1/3)(x+ y+ z)= c/3 when x= y= z= c/3, it cannot be more for any other values of x, y, and z.
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  3. #3
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    Re: Langrangean: Simple proof question, answer completely eluding me

    Hi there! Thank you so much for clearing that up for me!
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