What steps would I take or use in order to

therightwomanwf 2022-07-12 Answered
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?
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Answers (2)

Asdrubali2r
Answered 2022-07-13 Author has 14 answers
EDIT

Recall 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 > 0.

We also have that f ( 1 ) = cos ( 1 ) 1. But cos ( x ) < 1, x 2 n π, where n Z . Clearly, 1 2 n π, where n Z . Hence, we have that cos ( 1 ) < 1 f ( 1 ) < 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 < 0. ( a = 0, b = 1, f ( a ) = 1 and f ( b ) = cos ( 1 ) 1 < 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].

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nidantasnu
Answered 2022-07-14 Author has 7 answers
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.

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