Let g ≥ 0 on [ a , ∞ ), and ∫ a ∞ g ( x ) d x &lt; ∞. Also, f is continuous, bounded on [ a , ∞ ). Show that there exists a ξ ∈ [ a , ∞ ) such that ∫ a ∞ f ( x ) g ( x ) d x = f ( ξ ) ∫ a ∞ g ( x ) d x .This is extension of Intermediate value theorem.

Question

Let   g  ≥  0 on   [  a  ,  ∞  ), and       ∫    a    ∞    g  (  x  )    d  x  &amp;lt;  ∞. Also,   f is continuous, bounded on   [  a  ,  ∞  ). Show that there exists a   ξ  ∈  [  a  ,  ∞  ) such that      ∫    a    ∞    f  (  x  )  g  (  x  )    d  x  =  f  (  ξ  )      ∫    a    ∞    g  (  x  )    d  x  .This is extension of Intermediate value theorem.

Carter Escobar · Accepted Answer

Let   M  =      sup          x      ≥      a        f  (  x  )  ,  m  =      inf          x      ≥      a        f  (  x  ) and   G  =      ∫          x      ≥      a        g  (  x  )  d  x. Since   g  (  x  )  ≥  0 we have   m  g  (  x  )  ≤  f  (  x  )  g  (  x  )  ≤  M  g  (  x  ) and so   m  G  ≤  I  =      ∫          x      ≥      a        f  (  x  )  g  (  x  )  d  x  ≤  M  G.If   G  =  0, then   g  (  x  )  =  0 ae. and hence   I  =  0 and so any   x  ≥  a will do, so suppose   G  &amp;gt;  0.If   m  G  =  I, then we must have   f  (  x  )  =  m ae., so we can choose any   x such that   f  (  x  )  =  m. Similarly for the upper bound.Hence we may assume that   m  &amp;lt;            I      G        &amp;lt;  M. By definition of inf,sup there exist       x    m    ,      x    M   such that   m  ≤  f  (      x    m    )  &amp;lt;            I      G        &amp;lt;  f  (      x    M    )  ≤  M and then by the usual intermediate value theorem we can find some   x  ∈  [      x    m    ,      x    M    ] such that   f  (  x  )  =            I      G       which is the desired result.

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