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

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Chapter 4, 4.2 - Problem 17 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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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.

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