`2x + cos(x) = 0` Show that the equation has exactly one real root.
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.