What steps would I take or use in order to

therightwomanwf

therightwomanwf

Answered question

2022-07-12

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?

Answer & Explanation

Asdrubali2r

Asdrubali2r

Beginner2022-07-13Added 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].
nidantasnu

nidantasnu

Beginner2022-07-14Added 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.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?