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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 statemnts 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(2) = 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?
Most Upvoted Answer
A function f(x) is continuous in the interval [0,2]. It is known that ...
Understanding the Problem:
We are given a function f(x) that is continuous in the interval [0,2]. We also know that f(0) = f(2) = -1 and f(1) = 1. Our task is to determine which of the given statements must be true based on this information.

Solution:
To solve this problem, we will analyze each statement one by one.

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

To prove this statement, we can use the Intermediate Value Theorem. According to this theorem, if a function is continuous on a closed interval [a, b] and takes on two different values, then it must also take on every value in between.

In this case, f(x) is continuous on the interval [0,2] and f(0) = f(2) = -1. Therefore, by the Intermediate Value Theorem, f(x) must take on every value between -1 in the interval [0,2].

Since f(1) = 1, there must exist a value y in the interval (0,1) such that f(y) = 1. Therefore, Statement A is true.

Statement B:
For every y in the interval (0,1), f(2) = f(2-y)

To disprove this statement, we can provide a counterexample. Let's assume y = 0.5. According to the given information, f(2) = -1 and f(1.5) is not equal to -1. Therefore, Statement B is false.

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

To determine the maximum value of the function, we need to find the critical points. Since f(x) is continuous on the interval [0,2], it must have a maximum value within this interval.

Since f(0) = f(2) = -1 and f(1) = 1, the function reaches a maximum value of 1 within the interval [0,2]. Therefore, Statement C is true.

Statement D:
There exists a y in the interval (0,1) such that f(y) = -f(2-y)

To prove this statement, we can substitute y = 0.5 into the equation.
f(0.5) = -f(2-0.5)
Since f(0.5) = 1 and f(1.5) = -1, the equation becomes:
1 = -(-1)
1 = 1

Therefore, Statement D is true.

Conclusion:
Based on our analysis, we have determined that Statement A and Statement D must be true.
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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 statemnts 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(2) = 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?
Question Description
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 statemnts 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(2) = 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]. It is known that f(0) = f(2) = -1 and f(1) =1 . which one of the following statemnts 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(2) = 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]. It is known that f(0) = f(2) = -1 and f(1) =1 . which one of the following statemnts 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(2) = 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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