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Thread: Inequality rational power

  1. #1
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    Inequality rational power

    Show that if a,b and c d are positive ( and c and d are rational), then $\displaystyle (a^c-b^c)(a^d-b^d) \geq 0$ Can we proceed by multiplying the factors like :

    $\displaystyle a^{c+d}-b^ca^d-a^cb^d+b^{c+d} \geq 0$ Can somebody brief me the concept behind this..... I will be greatful....
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  2. #2
    Junior Member Nehushtan's Avatar
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    Re: Inequality rational power

    Given $\displaystyle a,b\in\mathbb R$, either $\displaystyle a\geqslant b$ or $\displaystyle a\leqslant b$. Moreover, since $\displaystyle a,b,c,d$ are positive, $\displaystyle a\geqslant b$ $\displaystyle \implies$ $\displaystyle a^c\geqslant b^c$ and $\displaystyle a^d\geqslant b^d$; similarly for $\displaystyle a\leqslant b$. In both cases, $\displaystyle a^c-b^c$ and $\displaystyle a^d-b^d$ have the same sign (or are zero) and so their product is non-negative.
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