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Statement-1:For any n in N, we have int_(0)^(npi) |(sinx)/(x)|dx ge (2)/(pi)(1+(1)/(2)+(1)/(3)+....+(1)/(n)) Statement-2:(sin x)/(x)ge(2)/(pi)"on"(0,(pi)/(2)) |
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Answer» Statement-1 is true, Statement-2 is True,Statement-2 is a correct explanation for Statement-1. `underset(0)overset(NPI)int |(sinx)/(x)|DX` `rArrI=underset(r=1)overset(n)sum underset((r-1)pi)overset(rpi)int|(sinx)/(x)|dx` `rArr I=underset(r=1)overset(n)sum underset(0)overset(pi)int|(sin{(r=1)pi+u)/((r-1)pi+u)|du`, where `x=(r-1)pi+u` `rArr I=underset(r=1)overset(n)sum underset0)overset(pi)int(SINU)/((r=1)pi+u)du` `rArr I ge underset(r=1)overset(n)sum underset(0)overset(pi)int(sin u)/((r=2)pi+pi)du[:'(sin u)/((r-1)pi+u)gt(sinu)/((r-1)pi+pi)]` `rArr I ge underset(r=2)overset(n)sum(2)/(r pi)rArr (2)/(pi)(1+(1)/(2)+...+(1)/(n))` So, statement-1 is true. Let f(x)`=(sinx)/(x)`. Then, `f'(x)=(xcosx-sinx)/(x^(2))=(g(x))/(x^(2))` where g(x)=x cosx-sinx Now, `g'(x)=-xsinx lt 0` for all`x in (0,pi//2)` `rArr g(x)` is decreasing on `xin (0,pi//2)` `rArr g(x) lt g(0)" for all "x in (0,pi//2)` `rArr f'(x) lt o" for all "x in(0,pi//2)` `:.f'(x) lt 0" for all "x in(0,pi//2)` `rArr f(x)` is decreasing on `(0,pi//2)` `rArr f(x) gt f(pi//2) " for all "x in(0,pi//2)` `rArr (sinx)/(x)gt(2)/(pi)" for all "x in(0,pi//2)` So, statement-2 is true. But, it is not a correct explanation for statement-1. |
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