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Question Number 96607 by bemath last updated on 03/Jun/20

$$\mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}\mathrm{f}\left(\mathrm{x}\right)\mathrm{is}\mathrm{a}\mathrm{function}$$$$\mathrm{defined}\mathrm{on}\mathbb{R},\mathrm{satisfying}\mathrm{f}\left(1\right)=1$$$$\mathrm{and}\mathrm{for}\mathrm{any}\mathrm{x}\in \mathbb{R},\mathrm{f}(\mathrm{x}+5)⩾\mathrm{f}\left(\mathrm{x}\right)+5$$$$\mathrm{and}\mathrm{f}(\mathrm{x}+1)⩽\mathrm{f}\left(\mathrm{x}\right)+1.\mathrm{If}\mathrm{g}\left(\mathrm{x}\right)=$$$$\mathrm{f}\left(\mathrm{x}\right)+1-\mathrm{x},\mathrm{then}\mathrm{g}\left(2002\right)=\mathrm{\_}\mathrm{\_}\mathrm{\_}$$

Commented by bobhans last updated on 03/Jun/20

$$\mathrm{we}\mathrm{determine}\mathrm{f}\left(2002\right)\mathrm{first}.\mathrm{from}\mathrm{the}$$$$\mathrm{condition}\mathrm{given},\mathrm{we}\mathrm{have}\mathrm{f}\left(\mathrm{x}\right)+5⩽\mathrm{f}(\mathrm{x}+5)⩽\mathrm{f}(\mathrm{x}+4)+1$$$$⩽\mathrm{f}(\mathrm{x}+3)+2⩽\mathrm{f}(\mathrm{x}+2)+3$$$$⩽\mathrm{f}(\mathrm{x}+1)+4⩽\mathrm{f}(\mathrm{x}+5)$$$$\mathrm{Thus}\mathrm{the}\mathrm{equality}\mathrm{holds}\mathrm{for}\mathrm{all}.\mathrm{so}\mathrm{we}$$$$\mathrm{have}\mathrm{f}(\mathrm{x}+1)=\mathrm{f}\left(\mathrm{x}\right)+1.\mathrm{Hence}\mathrm{from}\mathrm{f}\left(1\right)=1$$$$\mathrm{we}\mathrm{get}\mathrm{f}\left(2\right)=2,\mathrm{f}\left(3\right)=3,...,\mathrm{f}\left(2002\right)=2002$$$$\mathrm{therefore}\mathrm{g}\left(2002\right)=\mathrm{f}\left(2002\right)+1-2002=1$$

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