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Let f be a strictly monotonic continuous real valued function defined on [a,b] such that f(a) < a and f(b) >b Then which one of the following is TRUE?
- a)There exists exactly one (c) ∈ (a,b) such that f(c) = 0
- b)There exist exactly two points c1 c2 ∈ (a,b) such that f(ci) = ci , i =1,2
- c)There exists no (c) ∈ (a,b) such that f(c) = c
- d)There exist infinitely many points (c) ∈ (a,b) such that f(c) = c
Correct answer is option 'A'. Can you explain this answer?
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Let f be a strictly monotonic continuous real valued function defined ...
Answer:
To prove that option A is true, we will use the Intermediate Value Theorem.
Intermediate Value Theorem:
If a function f is continuous on a closed interval [a, b] and f(a) and f(b) have opposite signs, then there exists at least one c in the interval (a, b) such that f(c) = 0.
Now let's consider the given function f(x) which is strictly monotonic (either strictly increasing or strictly decreasing) and continuous on the closed interval [a, b]. We are given that f(a) < a="" and="" f(b)="" /> b, which means that f(a) and f(b) have opposite signs.
Proof:
Let's assume that there exists more than one point c in the interval (a, b) such that f(c) = 0. Since f is strictly monotonic, this implies that f is either strictly increasing or strictly decreasing.
Case 1: f is strictly increasing:
If f is strictly increasing, then f(a) < f(c)="" />< f(b)="" for="" all="" c="" in="" the="" interval="" (a,="" b).="" but="" this="" contradicts="" the="" fact="" that="" f(a)="" />< a="" and="" f(b)="" /> b, as f(c) cannot be equal to 0.
Case 2: f is strictly decreasing:
If f is strictly decreasing, then f(a) > f(c) > f(b) for all c in the interval (a, b). Again, this contradicts the fact that f(a) < a="" and="" f(b)="" /> b, as f(c) cannot be equal to 0.
Therefore, in both cases, we have reached a contradiction. Hence, our assumption that there exists more than one point c such that f(c) = 0 is false.
Thus, there exists exactly one point c in the interval (a, b) such that f(c) = 0. Therefore, option A is true.
To prove that option A is true, we will use the Intermediate Value Theorem.
Intermediate Value Theorem:
If a function f is continuous on a closed interval [a, b] and f(a) and f(b) have opposite signs, then there exists at least one c in the interval (a, b) such that f(c) = 0.
Now let's consider the given function f(x) which is strictly monotonic (either strictly increasing or strictly decreasing) and continuous on the closed interval [a, b]. We are given that f(a) < a="" and="" f(b)="" /> b, which means that f(a) and f(b) have opposite signs.
Proof:
Let's assume that there exists more than one point c in the interval (a, b) such that f(c) = 0. Since f is strictly monotonic, this implies that f is either strictly increasing or strictly decreasing.
Case 1: f is strictly increasing:
If f is strictly increasing, then f(a) < f(c)="" />< f(b)="" for="" all="" c="" in="" the="" interval="" (a,="" b).="" but="" this="" contradicts="" the="" fact="" that="" f(a)="" />< a="" and="" f(b)="" /> b, as f(c) cannot be equal to 0.
Case 2: f is strictly decreasing:
If f is strictly decreasing, then f(a) > f(c) > f(b) for all c in the interval (a, b). Again, this contradicts the fact that f(a) < a="" and="" f(b)="" /> b, as f(c) cannot be equal to 0.
Therefore, in both cases, we have reached a contradiction. Hence, our assumption that there exists more than one point c such that f(c) = 0 is false.
Thus, there exists exactly one point c in the interval (a, b) such that f(c) = 0. Therefore, option A is true.
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Let f be a strictly monotonic continuous real valued function defined on [a,b]such that f(a)< aand f(b) >b Then which one of the following is TRUE?a)There exists exactly one (c) ∈ (a,b) such that f(c) = 0b)There exist exactly two points c1c2∈ (a,b) such that f(ci) = ci, i =1,2c)There exists no(c) ∈ (a,b) such that f(c) = cd)There exist infinitely many points (c) ∈ (a,b) such thatf(c) = cCorrect answer is option 'A'. Can you explain this answer?
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Let f be a strictly monotonic continuous real valued function defined on [a,b]such that f(a)< aand f(b) >b Then which one of the following is TRUE?a)There exists exactly one (c) ∈ (a,b) such that f(c) = 0b)There exist exactly two points c1c2∈ (a,b) such that f(ci) = ci, i =1,2c)There exists no(c) ∈ (a,b) such that f(c) = cd)There exist infinitely many points (c) ∈ (a,b) such thatf(c) = cCorrect answer is option 'A'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let f be a strictly monotonic continuous real valued function defined on [a,b]such that f(a)< aand f(b) >b Then which one of the following is TRUE?a)There exists exactly one (c) ∈ (a,b) such that f(c) = 0b)There exist exactly two points c1c2∈ (a,b) such that f(ci) = ci, i =1,2c)There exists no(c) ∈ (a,b) such that f(c) = cd)There exist infinitely many points (c) ∈ (a,b) such thatf(c) = cCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f be a strictly monotonic continuous real valued function defined on [a,b]such that f(a)< aand f(b) >b Then which one of the following is TRUE?a)There exists exactly one (c) ∈ (a,b) such that f(c) = 0b)There exist exactly two points c1c2∈ (a,b) such that f(ci) = ci, i =1,2c)There exists no(c) ∈ (a,b) such that f(c) = cd)There exist infinitely many points (c) ∈ (a,b) such thatf(c) = cCorrect answer is option 'A'. Can you explain this answer?.
Let f be a strictly monotonic continuous real valued function defined on [a,b]such that f(a)< aand f(b) >b Then which one of the following is TRUE?a)There exists exactly one (c) ∈ (a,b) such that f(c) = 0b)There exist exactly two points c1c2∈ (a,b) such that f(ci) = ci, i =1,2c)There exists no(c) ∈ (a,b) such that f(c) = cd)There exist infinitely many points (c) ∈ (a,b) such thatf(c) = cCorrect answer is option 'A'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let f be a strictly monotonic continuous real valued function defined on [a,b]such that f(a)< aand f(b) >b Then which one of the following is TRUE?a)There exists exactly one (c) ∈ (a,b) such that f(c) = 0b)There exist exactly two points c1c2∈ (a,b) such that f(ci) = ci, i =1,2c)There exists no(c) ∈ (a,b) such that f(c) = cd)There exist infinitely many points (c) ∈ (a,b) such thatf(c) = cCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f be a strictly monotonic continuous real valued function defined on [a,b]such that f(a)< aand f(b) >b Then which one of the following is TRUE?a)There exists exactly one (c) ∈ (a,b) such that f(c) = 0b)There exist exactly two points c1c2∈ (a,b) such that f(ci) = ci, i =1,2c)There exists no(c) ∈ (a,b) such that f(c) = cd)There exist infinitely many points (c) ∈ (a,b) such thatf(c) = cCorrect answer is option 'A'. Can you explain this answer?.
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