Show that there is a number x ∈ [ π / 2 , π ]...

stratsticks57jl

stratsticks57jl

Answered

2022-07-20

Show that there is a number x [ π / 2 , π ] such that tan ( x ) = x
Can I solve this by the intermediate value theorem?

Answer & Explanation

Wayne Everett

Wayne Everett

Expert

2022-07-21Added 19 answers

Yes, you can. Consider f ( x ) = tan x + x. Notice that f ( π / 2 + ϵ ) for small ϵ (more formally, the right limit as x π / 2 of f ( x ) is ) while f ( π ) = π. Now the intermediate value theorem gives you want you want.
Donna Flynn

Donna Flynn

Expert

2022-07-22Added 3 answers

Your problem is equivalent to proving that
(1) f ( z ) = z cot z = z 2 + π z 2 = g ( z )
for some z I = ( 0 , π 2 ) . That is trivial since f ( z ) decreases from 1 to 0 on I while g ( z ) increases from 0 to π 2 2 on I, and both f ( z ) and g ( z ) are continuous over I. By approximating z cot z with 1 z 2 3 we also get:
z 3 π + 192 + 9 π 2 16
hence the solution to the original problem is about:
(2) x 5 π + 192 + 9 π 2 16 .

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

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

Didn't find what you were looking for?