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Let : R -> [0, ∞) be a continuous functions. Then which one of the following is NOT TRUE?
(a) There exists x \in R such that f(x) = (f(0) + f(1))/2
(b) There exists x \in R such that f(x) = sqrt(f(- 1) * f(1))
(c) There exists x \in R such that f(x) = integrate f(t) dt from - 1 to 1
(d) There exists such that f(x) = integrate f(t) dt from 0 to 1 x \in R?
Most Upvoted Answer
Let : R -> [0, ∞) be a continuous functions. Then which one of the fol...
Understanding the Statements
To determine which statement is NOT TRUE regarding the continuous function f: R -> [0, ∞), let's analyze each option.
Statement (a)
- This statement claims that there exists an x in R such that f(x) equals the average of f(0) and f(1).
- By the Intermediate Value Theorem, since f is continuous, it will take every value between f(0) and f(1). Thus, this statement is TRUE.
Statement (b)
- This states that there exists an x in R such that f(x) equals the geometric mean of f(-1) and f(1).
- Similar to (a), the continuity of f implies that it will take on all values between f(-1) and f(1). Hence, this statement is also TRUE.
Statement (c)
- This asserts the existence of x in R such that f(x) equals the integral of f(t) dt from -1 to 1.
- The integral represents an area under the curve, and by the properties of continuous functions, there will exist a point in R where f equals the average value of the function over the interval. Thus, this statement is TRUE.
Statement (d)
- This states that there exists an x in R such that f(x) equals the integral of f(t) dt from 0 to 1.
- However, the integral from 0 to 1 may not necessarily equal any value that f takes on R, especially if f is not bounded or behaves irregularly in that interval. Thus, this statement is NOT necessarily TRUE.
Conclusion
- The statement that is NOT TRUE is (d).
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Let : R -> [0, ∞) be a continuous functions. Then which one of the following is NOT TRUE?(a) There exists x \in R such that f(x) = (f(0) + f(1))/2(b) There exists x \in R such that f(x) = sqrt(f(- 1) * f(1))(c) There exists x \in R such that f(x) = integrate f(t) dt from - 1 to 1(d) There exists such that f(x) = integrate f(t) dt from 0 to 1 x \in R?
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Let : R -> [0, ∞) be a continuous functions. Then which one of the following is NOT TRUE?(a) There exists x \in R such that f(x) = (f(0) + f(1))/2(b) There exists x \in R such that f(x) = sqrt(f(- 1) * f(1))(c) There exists x \in R such that f(x) = integrate f(t) dt from - 1 to 1(d) There exists such that f(x) = integrate f(t) dt from 0 to 1 x \in R? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let : R -> [0, ∞) be a continuous functions. Then which one of the following is NOT TRUE?(a) There exists x \in R such that f(x) = (f(0) + f(1))/2(b) There exists x \in R such that f(x) = sqrt(f(- 1) * f(1))(c) There exists x \in R such that f(x) = integrate f(t) dt from - 1 to 1(d) There exists such that f(x) = integrate f(t) dt from 0 to 1 x \in R? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let : R -> [0, ∞) be a continuous functions. Then which one of the following is NOT TRUE?(a) There exists x \in R such that f(x) = (f(0) + f(1))/2(b) There exists x \in R such that f(x) = sqrt(f(- 1) * f(1))(c) There exists x \in R such that f(x) = integrate f(t) dt from - 1 to 1(d) There exists such that f(x) = integrate f(t) dt from 0 to 1 x \in R?.
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