Abstract Algebra - Prove that f(c)=g(c) for some c ∈ [a,b]. Please help!?!?!
Favorites|Homepage
Subscriptions | sitemap
HOME > > Abstract Algebra - Prove that f(c)=g(c) for some c ∈ [a,b]. Please help!?!?!

Abstract Algebra - Prove that f(c)=g(c) for some c ∈ [a,b]. Please help!?!?!

[From: ] [author: ] [Date: 12-11-19] [Hit: ]
Prove that f(c) = g(c) for some c ∈ [a,b].I would appreciate your help. Thanks in advance for your help and time.-Consider the function h:[a,b]->R given by h(x) = f(x) - g(x).......
Suppose that f: [a,b] ---> R and g: [a,b]---> R are continuous functions such that f(a) ≤ g(a) and g(b) ≤ f(b). Prove that f(c) = g(c) for some c ∈ [a,b].

I would appreciate your help. Thanks in advance for your help and time.

-
Consider the function h:[a,b]->R given by h(x) = f(x) - g(x).
Since f and g are continuous functions, h must also be continuous which means the intermediate value theorem holds.
Now, if f(a) = g(a) or f(b) = g(b), then set c = a or c = b and we're done. Otherwise, we note that
h(a) < 0 and
h(b) > 0
by the intermediate value theorem, there must be a c in [a,b] such that h(c) = 0. If h(c) = 0, the f(c) = g(c).
In any case, there exists a c such that f(c) = g(c)
1
keywords: isin,help,Algebra,that,some,Please,Prove,for,Abstract,Abstract Algebra - Prove that f(c)=g(c) for some c ∈ [a,b]. Please help!?!?!
New
Hot
© 2008-2010 http://www.science-mathematics.com . Program by zplan cms. Theme by wukong .