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Let f(x) = (1-x)^(2) sin^(2)x+ x^(2) for...

Let `f(x) = (1-x)^(2) sin^(2)x+ x^(2)` for all `x in IR` and let `g(x) = int_(1)^(x)((2(t-1))/(t+1)-lnt) f(t)` dt for all `x in (1,oo)`.
Consider the statements :
P : There exists some `x in IR` such that `f(x) + 2x = 2 (1+x^(2))`
Q : There exist some `x in IR` such that `2f(x) + 1 = 2x(1+x)`
Then

A

both P and Q are true

B

P is true and Q is false

C

P is false and Q is true

D

both P and Q are false

Text Solution

Verified by Experts

The correct Answer is:
C
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RESONANCE-DEFINITE INTEGRATION & ITS APPLICATION -Exercise 3 Part - I
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  3. Let f(x) = (1-x)^(2) sin^(2)x+ x^(2) for all x in IR and let g(x) = in...

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  4. If f(x) = int(0)^(x) e^(t^(2)) (t-2) (t-3) dt for all x in (0, oo) , t...

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  6. Let f : [1/2, 1] rarr R (the set of the all real numbers) be a positiv...

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  7. For a in R (the set of all real numbers), a ne = - 1, lim(nrarroo)((...

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