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Math Help - inequalities

  1. #1
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    inequalities

    0≤a<b, prove that a^2≤ab<b^2. Also, show by example that it does not follow that a^2<ab<b^2
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  2. #2
    Super Member 11rdc11's Avatar
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    For the example assume a = 0 so b>0

    0^2<0<b^2

    which is false
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  3. #3
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    Quote Originally Posted by rmpatel5 View Post
    0≤a<b, prove that a^2≤ab<b^2. Also, show by example that it does not follow that a^2<ab<b^2
    a<b ........................eqn(1)

    Multiply both sides with a, we get,

    a^2<ab .....................eqn(2)

    Now multiply both sides of eqn(1) with b, we get

    ab<b^2 ...................eqn (3)

    from eqn(2) and eqn (3)

    a^2<ab<b^2
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  4. #4
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    Multiply both sides by a
    a<b \Rightarrow a^{2}\le ab since a can be 0

    Multiply both sides by b
    a<b \Rightarrow ab<b^{2}

    Hence a^{2}\le ab<b^{2}

    A counter example for a^{2}<ab<b^{2} is given by 11rdc11
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