Use the intermediate value theorem to show that the equation, tan(x)=2x has an infinite amount of real solutions.

Mariah Sparks 2022-07-19 Answered
Use the intermediate value theorem to show that the equation,
t a n ( x ) = 2 x
has an infinite amount of real solutions.
So far I have used the IVT to show that for f ( x ) = t a n ( x ) in the interval ( π 2 , π 2 ) there is a L between and such that there is a value c where f ( c ) = L
I also have shown that for g ( x ) = 2 x in the interval [ π 2 , π 2 ] there is a K between π and π such that there is a value d where f ( d ) = K
Would it be correct to say that because each of these functions have a value for each point in the interval and that the range of f ( x ) consists of all reals that f ( x ) must intersect with g ( x ) or there is some value x in the interval such that 2 x = t a n ( x )? If this is correct then how would I extend this result to all intervals [ π ( n 1 2 ) , π ( n + 1 2 ) ]?
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Answers (1)

Clarissa Adkins
Answered 2022-07-20 Author has 16 answers
Consider f ( x ) = tan ( x ) 2 x. Then f ( x ) = 1 cos 2 ( x ) 2.
On π 4 , π 2 and π 2 , π 4 , f ( x ) > 0.
Since lim x π 2 f ( x ) = + , lim x π 2 f ( x ) = + .
Similarly since lim x π 2 + f ( x ) = , lim x π 2 + f ( x ) = .
So x 1 ( π 2 , π 4 ), such that, f ( x 1 ) < 0, and x 2 ( π 4 , π 2 ), such that f ( x 2 ) > 0. By intermediate value theorem, x 0 , x 1 < x 0 < x 2 , f ( x 0 ) = 0 since f ( x ) is continuous.
Same reasoning can be applied to π 2 + k π , π 2 + k π, k Z . So there are infinite real solution for f ( x ).

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