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 = 0I think it can be solved using the intermediate value theorem but I can't find a suitable function.

Question

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  =  0I think it can be solved using the intermediate value theorem but I can&#039;t find a suitable function.

Cristopher Barrera · Accepted Answer

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  ].

Extrakt04 · Accepted Answer

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.

If f : [ a , b ] → <mrow class="MJX-TeXAtom-ORD">

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