Let I=[a, b] and let f,g be continuous on I and such that ∫ (a to b) f=∫(a to b) g. Prove that there exists c in I such that f(c)=g(c)

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embizze | High School Teacher | (Level 2) Educator Emeritus

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Given `int_a^bf(x)dx = int_a^b g(x)dx` prove `EEc in [a,b] ` such that `f(c)=g(c)` :

Assume that `f(c)!= g(c) AA cin[a,b]` . Then either `f(x)>g(x) AA x in [a,b]` or `g(x)>f(x) AA x in [a,b]` .

If `f(x)>g(x) AA x in [a,b]` then `int_a^bf(x)> int_a^b g(x)` which contradicts the given. So `f(x)<=g(x)` for some `x in[a,b]` .Simarly  `g(x)>f(x) AA x in [a,b] => int_a^bg(x)>int_a^bf(x)` which also contradicts the given.

Since f and g are continuous, there must exist a point c in [a,b] such that f(c)=g(c).

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