I received a question on a previous exam, but I had no clue how to go about doing it. I know I'm supposed to use the MVT, IVT and FTC, but I'm not sure where. The question is Suppose f(x) is integrable on [a,b], with f(x)>=0 on [a,b], and that g(x) is continuous on [a,b]. Assuming that f(x)g(x) is integrable on [a,b], show that EEc in [a,b]

Ivan Buckley 2022-09-17 Answered
I received a question on a previous exam, but I had no clue how to go about doing it. I know I'm supposed to use the MVT, IVT and FTC, but I'm not sure where. The question is
Suppose f ( x ) is integrable on [ a , b ], with f ( x ) 0 on [ a , b ], and that g ( x ) is continuous on [ a , b ]. Assuming that f ( x ) g ( x ) is integrable on [ a , b ], show that c [ a , b ] so that
a b f ( x ) g ( x ) d x = g ( c ) a b f ( x ) d x .
Thank you in advance.
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Answers (1)

Marley Stone
Answered 2022-09-18 Author has 13 answers
g ( x ) takes its maximum and minimum values M and m on [ a , b ]. Say, g ( x 1 ) = M and g ( x 2 ) = m. Then
g ( x 1 ) a b f ( x ) d x = M a b f ( x ) d x a b f ( x ) g ( x ) d x m a b f ( x ) d x = g ( x 2 ) a b f ( x ) d x .
Can you now see why there is a c [ a , b ] with
a b f ( x ) g ( x ) d x = g ( c ) a b f ( x ) d x ?

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