Prove that f(x) = x − a cos (x) − b has at least one real root for ∀a,b ∈ R


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Question Number 70742 by Joel122 last updated on 07/Oct/19

$Prove that f (x) = x - a \cos (x) - b$ $has at least one real root for \forall a, b \in R$
Commented by kaivan.ahmadi last updated on 07/Oct/19

$x - a c o s x - b = 0 \Rightarrow x - b = a c o s x$ $y_{1} = x - b$ $y_{2} = a c o s x$ $i f w e p l o t y_{1}, y_{2} t h e y c u t e a c h o t h e r a t l e a s t$ $i n o n e p o i n t . s o t h e e q u a l i t y y_{1} = y_{2} h a s a t l e a s t$ $o n e r o o t .$
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