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A function f(x) is continuous in the interval [0,2]  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 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)
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
A function f(x) is continuous in the interval [0,2] f(0) = f(2)= -1 an...
Let's define a new function g,
g(y) = f(y) -f(y+1)
Since function f is continuous in [0,2], therefore g would be continuous in [0,1] g(0) = -2, g(1) = 2
since g is continuous and goes from negative to positive value in [0,1]. therefore at some point g would be 0 in (0,1).
g=0 ⇒ f(y) = f(y+1) for some y in (0,1)
Therefore, correct answer would be (A).
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Most Upvoted Answer
A function f(x) is continuous in the interval [0,2] f(0) = f(2)= -1 an...
Explanation:

Given Information:
- Function f(x) is continuous in the interval [0,2]
- f(0) = f(2) = -1
- f(1) = 1

Statement Analysis:

a) There exists a y in the interval (0,1) such that f(y) = f(y+1)
- Since f(x) is continuous in the interval [0,2], by the Intermediate Value Theorem, there exists a y in (0,1) such that f(y) = f(y+1). This is a direct consequence of the continuity of the function.

b) For every y in the interval (0,1), f(y) = f(2-y)
- This statement cannot be guaranteed based on the given information as it is not specified that the function is symmetric about x = 1.

c) The maximum value of the function in the interval (0,2) is 1
- The function reaches a maximum value of 1 at x = 1, not necessarily the maximum value in the interval [0,2].

d) There exists a y in the interval (0,1) such that f(y) = -f(2-y)
- This statement is not necessarily true based on the given information. There is no direct relation between f(y) and f(2-y) specified.
Therefore, the correct statement that must be true is option 'a'.
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A function f(x) is continuous in the interval [0,2] f(0) = f(2)= -1 and f(1) = 1.Which one of thefollowing statements must be true?a)There exists a y in the interval (0,1) such that f(y) = f(y+1)b)For every in the interval (0,1), f(y) = f(2-y)c)The maximum value of the function in the interval (0,2) is 1d)There exists a y in the interval (0,1) such that f(y) = -f(2-y)Correct answer is option 'A'. Can you explain this answer?
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A function f(x) is continuous in the interval [0,2] f(0) = f(2)= -1 and f(1) = 1.Which one of thefollowing statements must be true?a)There exists a y in the interval (0,1) such that f(y) = f(y+1)b)For every in the interval (0,1), f(y) = f(2-y)c)The maximum value of the function in the interval (0,2) is 1d)There exists a y in the interval (0,1) such that f(y) = -f(2-y)Correct answer is option 'A'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about A function f(x) is continuous in the interval [0,2] f(0) = f(2)= -1 and f(1) = 1.Which one of thefollowing statements must be true?a)There exists a y in the interval (0,1) such that f(y) = f(y+1)b)For every in the interval (0,1), f(y) = f(2-y)c)The maximum value of the function in the interval (0,2) is 1d)There exists a y in the interval (0,1) such that f(y) = -f(2-y)Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A function f(x) is continuous in the interval [0,2] f(0) = f(2)= -1 and f(1) = 1.Which one of thefollowing statements must be true?a)There exists a y in the interval (0,1) such that f(y) = f(y+1)b)For every in the interval (0,1), f(y) = f(2-y)c)The maximum value of the function in the interval (0,2) is 1d)There exists a y in the interval (0,1) such that f(y) = -f(2-y)Correct answer is option 'A'. Can you explain this answer?.
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