Denote f(x)=2x+cos(x). f is continuous and diffetentiable everywhere. Also f is strictly monotone because f'(x)=2-sin(x)>0.

Also, f(-1)=cos(-1)-2<0 and f(0)=1>0. By the Intermediate Value Theorem there is a root of f on (-1, 0). And this root is unique because f is monotone.

## See

This Answer NowStart your **subscription** to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

Already a member? Log in here.

Denote f(x)=2x+cos(x). f is continuous and diffetentiable everywhere. Also f is strictly monotone because f'(x)=2-sin(x)>0.

Also, f(-1)=cos(-1)-2<0 and f(0)=1>0. By the Intermediate Value Theorem there is a root of f on (-1, 0). And this root is unique because f is monotone.