Prove that f ( x ) = 3 has a solution on the interval [ a , b ]

skottyrottenmf

skottyrottenmf

Answered question

2022-05-26

Prove that f ( x ) = 3 has a solution on the interval [ a , b ]

And Intermediate Value Theorem says that

if f ( a ) f ( b ) < 0, then it has a solution on that interval

So instead of seeing if 3 is between the interval and stuff like that. Can't I just do this:
f ( x ) = 3
f ( x ) 3 = 0
And then we consider f ( x ) 3 a completely new whole function called g ( x ) = 0

According to the theorem, I can say that since g ( a ) g ( b ) < 0, it has a solution on the interval.

Answer & Explanation

vard6vv

vard6vv

Beginner2022-05-27Added 12 answers

Yes, you can do this. There is a variant of this argument which allows you to prove the Brouwer fixed point theorem in one dimension: suppose f : [ a , b ] [ a , b ] is a continuous function. Show there exists a c [ a , b ] with f ( c ) = c.
It's a great exercise to try out, and from your question you are halfway there already.

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