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Let g (x) ` int_(0)^(x) f(t) dy=t` , where f is such that ` (1)/(2) le f (t) le 1` for ` tin [0,1] and 0 le f (t) le (1)/(2) ` for ` t in [1 ,2]`. then . g (2) satis fies the inequalityA. `-(3)/(2)leg (2) lt(1)/(2)`B. ` 0leg (2) lt2`C. `(3)/(2) lt g (2) le 5//2`D. `2lt g (2) lt 4` |
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Answer» Correct Answer - B Given , g (x) `= int_(0)^(x)f(t) dt` ` rArr g (2) = int_(0)^(2) f (t) dt = int_(0)^(1)f(t)dt + int_(1)^(2)f(t)dt` Now , `(1)/(2) le f (t) le 1 " for" t in [0,1]` We get `int_(0)^(1)(1)/(2)dt leint_(0)^(1)f(t)dt le int_(0)^(1)1 dt` `rArr (1)/(2)leint_(0)^(1)f (t) dt le 1` . . . (i) Again , `0le f (t) le (1)/(2) " for " t in [1,2]` . . . (ii) `rArr int_(1)^(2)0 dt le int_(1)^(2) f (t) dtle int _(1)^(2) dt` `rArr 0 le int_(1)^(2) f (t) dt le (1)/(2)` From Eqs . (i) and (ii) , we get `(1)/(2)le int_(0)^(1)f (t) dt + int_(1)^(2) f (t) dt le (3)/(2)` ` rArr (1)/(2) le g (2) le (3)/(2)` ` rArr 0leg (2) lt 2` |
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