Consider the equation 1 ? 2x = sin x. (a) Use the Intermediate Value Theorem to

Question

Consider the equation 1 ? 2x = sin x.
(a) Use the Intermediate Value Theorem to prove that this equation has a
root in the interval (0, pie/4). Check all the criterions in the IVT.
(b) Use a graphing calculator to find an interval of length 0.01 that contains
this root. Explain how to obtain this interval.

Explanation / Answer

(a)f(x)=sinx+2x-1 Notice that, f(0) < 0 and f(pi/4) > 0 Therefore, by the Intermediate Value theorem there exists a zero in the interval [0, pi/4]. Now, is this the only root? f'(x) = cos x+2 for all x. Therefore, f(x) is strictly increasing and therefore, the root we found above is the ONLY root.

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