Prove-that-f-x-x-a-cos-x-b-has-at-least-one-real-root-for-a-b-R-

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 .