20170927.1

题 1:

设 $0 < a < b$ ,证明:$\ln \dfrac {b} {a}>2\dfrac {b-a} {b+a}$.


证明:

$\ln \dfrac {b} {a}>\dfrac {\dfrac {b} {a}-1} {\dfrac {b} {a}+1}$ , 令 $\dfrac {b} {a}=x$ , 即证 $\ln x>2\dfrac {x-1} {x+1}$。

令 F(x) = (1+x)lnx-2(x-1) ,则:
$F’\left( x\right) =\dfrac {1} {x}+\ln x-1$
$F’’\left( x\right) =-\dfrac {1} {x^{2}}+\dfrac {1} {x}=\dfrac {1} {x}\left( 1-\dfrac {1} {x}\right) > 0$

因此 F’(x) > F’(1) = 0 ,故 F(x) >F(1)=0。得证。


THE END.

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