Use the intermediate value theorem to prove that if f : [ 1 , 2...

Barrett Osborn

Barrett Osborn

Answered

2022-11-23

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.

Answer & Explanation

Kailee Abbott

Kailee Abbott

Expert

2022-11-24Added 14 answers

Note that because f ( x ) is continuous on [ 1 , 2 ], the function f ( x ) is bounded on [ 1 , 2 ]. Suppose that | f ( x ) | < 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 > 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.

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