Use the intermediate value theorem to prove that if f : [ 1 , 2 ] → R is a continuous function, that there is at least one number c in the interval ( 1 , 2 ) such that f ( c ) = 1 / ( 1 − c ) + 1 / ( 2 − c )This is a question for my intro calc class that I am having a hard time understanding.

Question

Use the intermediate value theorem to prove that if   f  :  [  1  ,  2  ]  →  R is a continuous function, that there is at least one number   c in the interval   (  1  ,  2  ) such that   f  (  c  )  =  1      /    (  1  −  c  )  +  1      /    (  2  −  c  )This is a question for my intro calc class that I am having a hard time understanding.

Kailee Abbott · Accepted Answer

Note that because   f  (  x  ) is continuous on   [  1  ,  2  ], the function   f  (  x  ) is bounded on   [  1  ,  2  ]. Suppose that       |    f  (  x  )      |    &amp;lt;  B for all   x in our interval. Let  g  (  x  )  =  f  (  x  )  −      1          1      −      x        −      1          2      −      x        .There is an   a in (1,2) such that   g  (  a  ) is positive, and a   b such that   g  (  b  ) is negative, and hence by the Intermediate Value Theorem there is a   c between   a and   b such that   g  (  c  )  =  0.Detail: We show that there is indeed an   a such that   g  (  a  ) is positive.In order to have fewer minus signs, note that  g  (  x  )  =  f  (  x  )  +      1          x      −      1        −      1          2      −      x        .Note that       1          x      −      1       becomes very large positive for   x close enough to 1 but to the right of 1.The term       1          2      −      x       is close to 1 when   x is close to 1. So by choosing   a near 1 such that       1          a      −      1        &amp;gt;  B  +  2, we can make sure that   g  (  a  ) is positive. For then the   f  (  a  )  −      1          2      −      a       part cannot be negative enough to make   g  (  a  ) negative.For   b, we play the same game near 2. For   x near 2 but to the left of 2, the term       1          2      −      x       is large positive, so   g  (  x  ) is large negative.

Use the intermediate value theorem to prove that if f:[1,2]->R is a continuous function, that there is at least one number c in the interval (1,2) such that f(c)=1/(1−c)+1/(2−c) This is a question for my intro calc class that I am having a hard time understanding.

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