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Thread: 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

    $\displaystyle 0^2<0<b^2$

    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
    $\displaystyle a<b$ ........................eqn(1)

    Multiply both sides with a, we get,

    $\displaystyle a^2<ab$ .....................eqn(2)

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

    $\displaystyle ab<b^2$ ...................eqn (3)

    from eqn(2) and eqn (3)

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

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

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

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