Question Number 70742 by Joel122 last updated on 07/Oct/19
$$\mathrm{Prove}\:\mathrm{that}\:{f}\left({x}\right)\:=\:{x}\:−\:{a}\:\mathrm{cos}\:\left({x}\right)\:−\:{b} \\ $$$$\mathrm{has}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{real}\:\mathrm{root}\:\mathrm{for}\:\forall{a},{b}\:\in\:\mathbb{R} \\ $$
Commented by kaivan.ahmadi last updated on 07/Oct/19
$${x}−{acosx}−{b}=\mathrm{0}\Rightarrow{x}−{b}={acosx} \\ $$$${y}_{\mathrm{1}} ={x}−{b} \\ $$$${y}_{\mathrm{2}} ={acosx} \\ $$$${if}\:{we}\:{plot}\:{y}_{\mathrm{1}} ,{y}_{\mathrm{2}} \:{they}\:{cut}\:{each}\:{other}\:{at}\:{least} \\ $$$${in}\:{one}\:{point}.{so}\:{the}\:{equality}\:{y}_{\mathrm{1}} ={y}_{\mathrm{2}} \:{has}\:{at}\:{least} \\ $$$${one}\:{root}. \\ $$