Let f : [ a , b ] → R be continuous, p , q &gt; 0. We want to show that there is a c ∈ [ a , b ] s.t. p f ( a ) + q f ( b ) = ( p + q ) f ( c ). p p + q , q p + q are both in (0,1). Then p p + q f ( a ) + q p + q f ( b ) is in ( 0 , f ( a ) + f ( b ) ). If we want to apply the intermediate value theorem we better would need it to be in [ f ( u ), f ( w )], where we know that f ( u ) is the Minimum and f ( w ) is the Maximum. But we have not any info about what the Min/Max would be?

Question

Let   f  :  [  a  ,  b  ]  →      R   be continuous,   p  ,  q  &amp;gt;  0. We want to show that there is a   c  ∈  [  a  ,  b  ] s.t.   p  f  (  a  )  +  q  f  (  b  )  =  (  p  +  q  )  f  (  c  ).       p          p      +      q        ,      q          p      +      q       are both in (0,1). Then       p          p      +      q        f  (  a  )  +      q          p      +      q        f  (  b  ) is in   (  0  ,  f  (  a  )  +  f  (  b  )  ). If we want to apply the intermediate value theorem we better would need it to be in [  f  (  u  ),  f  (  w  )], where we know that   f  (  u  ) is the Minimum and   f  (  w  ) is the Maximum. But we have not any info about what the Min/Max would be?

Myla Pierce · Accepted Answer

p          p      +      q        f  (  a  )  +      q          p      +      q        f  (  b  ) is also in [  m,  M] where   m is the minimum and   M is the maximum. For example,   f  (  a  )  ≤  M and   f  (  b  )  ≤  M implies       p          p      +      q        f  (  a  )  +      q          p      +      q        f  (  b  )  ≤      p          p      +      q        M  +      q          p      +      q        M  =  M and similarly for the minimum.

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