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Let : R -> [0, ∞) be a continuous functions. Then which 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?
(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 of the followi...
Understanding the Problem
The question presents four statements regarding a continuous function f: R -> [0, ∞) and asks to identify which statement is not true.
Analyzing Each Statement
Conclusion
The statement that is not true is (c). While (a), (b), and (d) can be justified under certain conditions, (c) is not universally valid for all continuous functions.
The question presents four statements regarding a continuous function f: R -> [0, ∞) and asks to identify which statement is not true.
Analyzing Each Statement
- (a) There exists x ∈ R such that f(x) = (f(0) + f(1))/2
- By the Intermediate Value Theorem, since f is continuous, it must take on every value between f(0) and f(1). Hence, the average (f(0) + f(1))/2 is achievable.
- (b) There exists x ∈ R such that f(x) = sqrt(f(-1) * f(1))
- This statement can also be justified using the Intermediate Value Theorem. Consider values f(-1) and f(1), f(x) can achieve the geometric mean if f is continuous.
- (c) There exists x ∈ R such that f(x) = integrate f(t) dt from -1 to 1
- The integral from -1 to 1 gives a fixed value. This does not necessarily equal any value of f(x) unless the function is constant. Thus, it may not hold true for all continuous functions.
- (d) There exists x ∈ R such that f(x) = integrate f(t) dt from 0 to 1
- Similar to statement (c), this integral results in a specific value, but it could match some f(x) if f is constructed appropriately. However, it is not guaranteed for all continuous functions.
Conclusion
The statement that is not true is (c). While (a), (b), and (d) can be justified under certain conditions, (c) is not universally valid for all continuous functions.
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Let : R -> [0, ∞) be a continuous functions. Then which 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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