Use the Intermediate Value Theorem to show that cosx=x have at least a solution in [0,π]?

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Use the Intermediate Value Theorem to show that cosx=x have at least a solution in [0,π]?

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Answer 1 · 2018-04-13T18:53:54

We have:

$cos x - x = 0$ $cosx - x= 0$

Now let $y = cos x - x$ $y = cosx - x$ . We see that $y (0) = cos (0) - 0 = 1$ $y(0) = cos(0) - 0 = 1$ and $y (π) = - 1 - π$ $y(pi) = -1 - pi$

Since $y (π) < 0 < y (0)$ $y(pi) < 0 < y(0)$ , and $y$ $y$ is continuous, there must be a value of $x$ $x$ in $[0, π]$ $[0, pi]$ where $cos x - x = 0$ $cosx -x = 0$ .

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Use the Intermediate Value Theorem to show that cosx=x have at least a solution in [0,π]?

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