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Question Number 20625 by mondodotto@gmail.com last updated on 29/Aug/17

Commented by $@ty@m last updated on 30/Aug/17

$$itseemssomethingmissinginthe$$$$questionbecausein{}^{\prime }{if}^{\prime }part,there$$$$is{}^{\prime }>c^{\prime }butthereisno{}^{\prime }{c}^{\prime }{in}^{\prime }prove{that}^{\prime }part.$$

Answered by ajfour last updated on 30/Aug/17

$$letf\left(x\right)=\frac{{ae}^x+{be}^{-x}}{e^x+e^{-x}}=\frac{a(e^{2x}+b/a)}{e^{2x}+1}$$$$asb/a<1\Rightarrow f\left(x\right)<a$$$$Alsof\left(x\right)=\frac{{ae}^x+{be}^{-x}}{e^x+e^{-x}}=\frac{b(e^{-2x}+a/b)}{e^{-2x}+1}$$$$andasa>b\Rightarrow f\left(x\right)>b.$$$$henceb<f\left(x\right)<a.$$

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