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I need help proving this inequality.

$\int_{1}^{2} \left(\frac{x+1}{2} \right)^{x+1} \ dx < \int_{1}^{2} x^x \ dx$

Normally I would define $f(x) = \left(\displaystyle\frac{x+1}{2} \right)^{x+1}$ and $g(x) = x^x$ and try to compare those two.
Any suggestions?

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    Evaluate each i$n$tegrals separately. For f(x) and g(x), evaluate us$i$ng numer$i$c$a$l methods.2011-01-23

1 Answers 1

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HINT: In $[1,2]$ we have that $x^x > (\frac{x+1}{2})^{x+1}$. Try to show this by finding the minimum of $h(x) = x\log(x)-(x+1)\log(\frac{x+1}{2})$.

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    oh good point! thanks2011-01-24