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Jamiya Weber 2022-06-25 Answered
If f : [ a , b ] R is continuous on [ a , b ], show that c [ a , b ] so that
c a b c f ( x ) d x = 0
I think it can be solved using the intermediate value theorem but I can't find a suitable function.
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Answers (2)

Cristopher Barrera
Answered 2022-06-26 Author has 24 answers
If it makes sense, equivalently, c a , c b [ a , b ] (for example if [ a , b ] = [ 0 , 1 ]) for every c [ a , b ] define g ( x ) = c a b c f ( x ) d x, g ( a ) = g ( b )

Take c [ 50 , 51 ] for example, c 50 , 51 c [ 0 , 1 ].
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Extrakt04
Answered 2022-06-27 Author has 5 answers
As stated, is true even without the continuity hypothesis. Take b c = c a, i.e., c = ( a + b ) / 2 the midpoint of the interval.
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