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

[Sujith Sizon]

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 $f(y) = −f(2−y)$----------
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?

 

[pasmith]

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