Let f and g be continuous functions on [a,b] such that f(a)>=g(a) and f(b)<=g(b). Prove f(x_0)=g(x_0) for at least one x_0 in [a,b].

Leypoldon

Leypoldon

Answered question

2022-08-10

Let f and g be continuous functions on [ a , b ] such that f ( a ) g ( a ) and f ( b ) g ( b ). Prove f ( x 0 ) = g ( x 0 ) for at least one x 0 in [ a , b ].
Here's what I have so far:
Let h be a continuous function where h = f g
Since h ( b ) = f ( b ) g ( b ) 0, and h ( a ) = f ( a ) g ( a ) 0, then h ( b ) 0 h ( a ) is true.
So, by IVT there exists some y such that...
And that's where I need help. I read what the IVT is, but I could use some help explaining how it applies here, and why it finishes the proof. Thank you!

Answer & Explanation

beentjie8e

beentjie8e

Beginner2022-08-11Added 20 answers

There exists some y ( a , b ) such that h ( y ) = 0. So f ( y ) = g ( y ). The other case a = b is trivial.

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