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Let f: (0, 1) → R be a continuously differentiable function such that f' has finitely many zeros in (0, 1) and f' changes sign at exactly two of these points. Then for any y ∈ R , the maximum number of solutions to f(x) = y in (0, 1) is ______________
Correct answer is '3'. Can you explain this answer?
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Let f: (0, 1) R be a continuously differentiable function such that f...
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Given, a continuously differentiable function f: (0, 1) R such that it has finitely many zeros in (0, 1) and changes sign at exactly two of these points. We need to find the maximum number of solutions to f(x) = y in (0, 1) for any y R.
Case 1: y is not a value attained by f(x) in (0, 1)
In this case, the number of solutions to f(x) = y in (0, 1) is zero.
Case 2: y is a value attained by f(x) in (0, 1)
Let the two points where f(x) changes sign be a and b. Without loss of generality, assume that f(a) < y < f(b). Since f(x) is continuous, by the Intermediate Value Theorem, there exists at least one solution to f(x) = y in (a, b). Let c be a zero of f(x) in (0, a) or (b, 1), if it exists. Then, we have the following cases:
Therefore, the maximum number of solutions to f(x) = y in (0, 1) is 3.
Given, a continuously differentiable function f: (0, 1) R such that it has finitely many zeros in (0, 1) and changes sign at exactly two of these points. We need to find the maximum number of solutions to f(x) = y in (0, 1) for any y R.
Case 1: y is not a value attained by f(x) in (0, 1)
In this case, the number of solutions to f(x) = y in (0, 1) is zero.
Case 2: y is a value attained by f(x) in (0, 1)
Let the two points where f(x) changes sign be a and b. Without loss of generality, assume that f(a) < y < f(b). Since f(x) is continuous, by the Intermediate Value Theorem, there exists at least one solution to f(x) = y in (a, b). Let c be a zero of f(x) in (0, a) or (b, 1), if it exists. Then, we have the following cases:
- Case 2.1: f(c) > y. In this case, there can be at most one solution to f(x) = y in (c, a) or (b, c). Therefore, the maximum number of solutions is 2.
- Case 2.2: f(c) = y. In this case, there can be at most two solutions to f(x) = y in (c, a) or (b, c). Therefore, the maximum number of solutions is 3.
- Case 2.3: f(c) < y. In this case, there can be at most one solution to f(x) = y in (c, a) and at most one solution in (b, c). Therefore, the maximum number of solutions is 3.
Therefore, the maximum number of solutions to f(x) = y in (0, 1) is 3.
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Let f: (0, 1) R be a continuously differentiable function such that f has finitely many zeros in (0, 1) and fchanges sign at exactly two of these points. Then for any y R , the maximum number of solutions to f(x) = y in (0, 1) is ______________Correct answer is '3'. Can you explain this answer?
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Let f: (0, 1) R be a continuously differentiable function such that f has finitely many zeros in (0, 1) and fchanges sign at exactly two of these points. Then for any y R , the maximum number of solutions to f(x) = y in (0, 1) is ______________Correct answer is '3'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Let f: (0, 1) R be a continuously differentiable function such that f has finitely many zeros in (0, 1) and fchanges sign at exactly two of these points. Then for any y R , the maximum number of solutions to f(x) = y in (0, 1) is ______________Correct answer is '3'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f: (0, 1) R be a continuously differentiable function such that f has finitely many zeros in (0, 1) and fchanges sign at exactly two of these points. Then for any y R , the maximum number of solutions to f(x) = y in (0, 1) is ______________Correct answer is '3'. Can you explain this answer?.
Let f: (0, 1) R be a continuously differentiable function such that f has finitely many zeros in (0, 1) and fchanges sign at exactly two of these points. Then for any y R , the maximum number of solutions to f(x) = y in (0, 1) is ______________Correct answer is '3'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Let f: (0, 1) R be a continuously differentiable function such that f has finitely many zeros in (0, 1) and fchanges sign at exactly two of these points. Then for any y R , the maximum number of solutions to f(x) = y in (0, 1) is ______________Correct answer is '3'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f: (0, 1) R be a continuously differentiable function such that f has finitely many zeros in (0, 1) and fchanges sign at exactly two of these points. Then for any y R , the maximum number of solutions to f(x) = y in (0, 1) is ______________Correct answer is '3'. Can you explain this answer?.
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