What steps would I take or use in order to use the intermediate value theorem to show that cos ⁡ x = x has a solution between x = 0 and x = 1?

Question

What steps would I take or use in order to use the intermediate value theorem to show that   cos  ⁡  x  =  x has a solution between   x  =  0 and   x  =  1?

Kiana Cantu · Accepted Answer

EDITRecall the statement of the intermediate value theorem.Theorem If   f  (  x  ) is a real-valued continuous function on the interval [a,b], then given any   y  ∈  [  min  (  f  (  a  )  ,  f  (  b  )  )  ,  max  (  f  (  a  )  ,  f  (  b  )  )  ], there exists   c  ∈  [  a  ,  b  ] such that   f  (  c  )  =  y.The theorem guarantees us that given any value   y in-between   f  (  a  ) and   f  (  b  ), the continuous function   f  (  x  ) takes the value   y for some point in the interval [a,b].Now lets get back to our problem. Look at the function   f  (  x  )  =  cos  ⁡  (  x  )  −  x.We have   f  (  0  )  =  1  &amp;gt;  0.We also have that   f  (  1  )  =  cos  ⁡  (  1  )  −  1. But   cos  ⁡  (  x  )  &amp;lt;  1,   ∀  x  ≠  2  n  π, where   n  ∈      Z  . Clearly,   1  ≠  2  n  π, where   n  ∈      Z  . Hence, we have that   cos  ⁡  (  1  )  &amp;lt;  1    ⟹    f  (  1  )  &amp;lt;  0.Hence, we have a continuous function   f  (  x  )  =  cos  ⁡  (  x  )  −  x on the interval [0,1] with   f  (  0  )  =  1 and   f  (  1  )  =  cos  ⁡  (  1  )  −  1  &amp;lt;  0. (  a  =  0,   b  =  1,   f  (  a  )  =  1 and   f  (  b  )  =  cos  ⁡  (  1  )  −  1  &amp;lt;  0).Note that 0 lies in the interval   [  cos  ⁡  (  1  )  −  1  ,  1  ]. Hence, from the intermediate value theorem, there exists a   c  ∈  [  0  ,  1  ] such that f(c)=0.This means that c is a root of the equation. Hence, we have proved that there exists a root in the interval [0,1].

letumsnemesislh · Accepted Answer

You can apply the IVT to the continuous function   x      /    cos  ⁡  x to show that it takes on the value 1 for some   x,   0  ≤  x  ≤  1.

What steps would I take or use in order to

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