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Questions related to an unknown function given its values at some points

  1. Oct 10, 2015 #1

    Sujith Sizon

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    Gold Member

    A function ##f(x)## is continuous in the interval ##[0,2]##. It is known that ##f(0)=f(2)=−1## and ##f(1)=1##. Which one of the following statements must be true?

    (A) There exists a ##y## in the interval ##(0,1)## such that ##f(y)=f(y+1)##

    (B) For every ##y## in the interval ##(0,1),f(y) = f(2−y)##

    (C) The maximum value of the function in the interval ##(0,2)## is ##1##

    (D) There exists a ##y## in the interval ##(0,1)## such that ##f(y) = −f(2−y)##


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    Here's my approach:

    Consider a function ##g(y) = f(y) - f(y+1)## since ##f## is a continuous function and ##g## is a combination of ##f## so it is also continuous in ##[0,1]##.

    it is found that

    ##g(0)=f(0)-f(1)=-1-1=-2## and

    ##g(1)=f(1)-f(2)=1-(-1)=+2##

    since #g# goes from ##-2## to ##+2## and is continuous in ##(0,1)## therefore there has to be a point in b/w ##(0,1)## such that ##g(\text{that point}) = 0##

    when ##g(y)=0## for some ##$y\in(0,1)## then ##f(y) = f(y+1)##, Hence **statement A** is true.

    Using the same logic **statement D** is true too. It is clear that statement B and C are false.

    The answer booklet says that only **statement A** is true. There is no comment on **statement D**; should it be considered false? Is my approach okay?
     
  2. jcsd
  3. Oct 10, 2015 #2

    pasmith

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    Homework Helper

    I agree that D is also true. Sometimes question setters make mistakes.
     
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