74 votes 74 votes 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? There exists a $y$ in the interval $(0,1)$ such that $f(y) = f(y+1)$ For every $y$ in the interval $(0,1)$,$f(y)$ = $f(2-y)$ The maximum value of the function in the interval $(0,2)$ is $1$ There exists a $y$ in the interval $(0,1)$ such that $f(y)$ = $-f(2-y)$ Calculus gatecse-2014-set1 calculus continuity normal + – go_editor asked Sep 28, 2014 go_editor 21.0k views answer comment Share Follow See all 7 Comments See all 7 7 Comments reply Show 4 previous comments DebRC commented Nov 21, 2022 reply Follow Share @MANSI_SOMANI Think of the function f(x) graphically, 8 votes 8 votes Ray Tomlinson commented Oct 10, 2023 i edited by Ray Tomlinson Oct 10, 2023 reply Follow Share One Answer will clear every doubt https://gateoverflow.in/1925/gate-cse-2014-set-1-question-47?show=412885#a412885 1 votes 1 votes gvinay commented Nov 20, 2023 reply Follow Share watch this …..defentely you get solution...it is in bangali language but it is easy to understand… https://www.youtube.com/watch?v=G9LLmDv3EIs thank youu soo much. 0 votes 0 votes Please log in or register to add a comment.
0 votes 0 votes Option A is Correct but D is Wrong , Reason The difference between y and 2-y should be less than the length of the interval given . Difference between y and 2-y is 2 and length of interval is (0,1) Therefore D is Wrong pC answered Jan 14, 2016 pC comment Share Follow See all 11 Comments See all 11 11 Comments reply Arjun commented Jan 14, 2016 reply Follow Share Why is that needed? f is defined over [0,2] and so the difference can be 2. 0 votes 0 votes pC commented Jan 14, 2016 reply Follow Share In option d they mentioned the range (0,1) for y but actually range is coming as 2 . SO i believe this is wrong choice 0 votes 0 votes Arjun commented Jan 14, 2016 reply Follow Share not really, it just wants 'y' in (0,1). 1 votes 1 votes pC commented Jan 14, 2016 reply Follow Share I am not sure enough . Agreeing with you @arjun sir . 0 votes 0 votes Shreya Roy commented Aug 25, 2016 reply Follow Share Arjun Sir, -f(2-y) is continuous between -2 and -4 .. [reflect about y axis then horizontal left shift by 2 then reflect about x axis to get -f(2-y)from f(y)].. so we can't say f(y)will intersect with -f(2-y) in between 0 and 1 ie. f(y)=-f(2-y) for some y in the interval 0 to 1 . so d is false 0 votes 0 votes Arjun commented Aug 25, 2016 reply Follow Share That is D option rt? 0 votes 0 votes Shreya Roy commented Aug 25, 2016 i edited by Shreya Roy Aug 26, 2016 reply Follow Share Sorry , D is right bcz -f(2-y) is continuous between 0 and 2 ,I did the mistake cz I was shifting it left by 2 unit but it will be shifted towards right by 2 unit since shifting is done wrt. -y not respect to +y ,here I committed the mistake.. Yes both a and d correct 1 votes 1 votes Kaluti commented Jul 28, 2017 reply Follow Share by applying intermediate value theorem both options a and d are coming true 1 votes 1 votes Xylene commented Jul 29, 2017 reply Follow Share What is that theorem ? How to apply it here? Can you please explain ? @Kaluti 0 votes 0 votes bharti commented Aug 27, 2017 i edited by bharti Aug 27, 2017 reply Follow Share @xylene the Theorem is if g (x) is some function which is defined over a close interval(a,b) than the value of resultant function will always pass through (a,b). http://mathworld.wolfram.com/IntermediateValueTheorem.html 1 votes 1 votes Prasanna Kapil commented Oct 23, 2017 i edited by Prasanna Kapil Oct 23, 2017 reply Follow Share i think A & D both are correct as explained by Shreya 0 votes 0 votes Please log in or register to add a comment.
0 votes 0 votes This Answer contains the all concept in which students having doubt and the usefull comments are also included so that the correct answer is option A and D. Ray Tomlinson answered Oct 10, 2023 Ray Tomlinson comment Share Follow See all 0 reply Please log in or register to add a comment.
0 votes 0 votes One of better solution you can find from here..it is in english+bengali language but easy to understand. https://www.youtube.com/watch?v=G9LLmDv3EIs thank you soo much gvinay answered Nov 20, 2023 gvinay comment Share Follow See all 0 reply Please log in or register to add a comment.