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Math Help - Using inequality

  1. #1
    Senior Member pankaj's Avatar
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    Using inequality

    If a,b,c are positive real numbers such that a+b+c=1 and
    \lambda=min\{a^3+a^2bc,b^3+ab^2c,c^3+abc^2\},then find the value of \lambda
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  2. #2
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    Quote Originally Posted by pankaj View Post
    If a,b,c are positive real numbers such that a+b+c=1 and
    \lambda=min\{a^3+a^2bc,b^3+ab^2c,c^3+abc^2\},then find the value of \lambda
    I would like to know if this is the question flat out or if this question arose from another question...

    Without loss of generality, we can consider only a^3+a^2bc since the other two expressions are the same but with a=b and a=c, because there is no reason why you can't call the 3 numbers a,b,c in any order when you find them

    Now, if we let a approach zero, the expression approaches zero. From what you've said a cannot be 0, but you can let a keep getting smaller and the value of lamda will continue to decrease

    For instance, let b=.49998 and c=.49998 and see what happens

    Then let b=.4999999998 and c=.4999999998 and see what happens (with a calculator you'll probably get an answer of 0 with round off)
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