Prove using the IVT: For all positive integers k, cos x=x^k.

Nasir Sullivan

Nasir Sullivan

Answered question

2022-09-26

Applying IV theorem to practical problems
Prove using the IVT: For all positive integers k, c o s x = x k has a solution
I know that the IV states at an interval [a,b] some value c is bound to be hit atleast once, assuming that [a,b] is continuous(Please improve my explanation where needed).
So, using this logic, assuming that 0 = x k c o s x, I have to find values of x where this would be true?
f ( 0 ) = 0 1
f ( π / 2 ) = ( π / 2 ) k 0
f ( π / 2 ) > 0
By the IVT, there should be a value C inbetween [ 0 , π / 2 ] if I am correct?
Please correct me where possible. I think I understand the theorem but applying it to a question like this makes me believe otherwise.

Answer & Explanation

Marnovdk

Marnovdk

Beginner2022-09-27Added 6 answers

Step 1
Your interpretation is correct. As to writing style, you should give a definition of f(x) before discussing its properties.
Step 2
The derivative of f ( x ) = x k cos x is f ( x ) = k x k 1 + sin x ,, which is positive for x > 0 for any positive k. So f(x) is strictly increasing, and continuous, for x 0 and k > 0.. And so, since f ( 0 ) < 0 < f ( π / 2 ) , there is a unique x ( 0 , π / 2 ) such that f ( x ) = 0..

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