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?(adsbygoogle = window.adsbygoogle || []).push({});

(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?

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