Statement-2`f(x)=(sin x)/(x) lt 1 for lt x lt pi/2`
Statement -2 `f(x)=(sin x)/(x)` is decreasing function on `(0,pi//2)`

Options

A. Statement-1 True statement -1 is True,Statement -2 is True statement -2 is a correct explanation for Statement-4
B. Statement-1 True statement -1 is True,Statement -2 is True statement -2 is not a correct explanation for Statement-4
C. Statement-1 True statement -1 is True,Statement -2 is False
D. Statement-1 is False ,Statement -2 is True

Answer

A

Explanation

We have `f(x)=(sinx)/(x)`
`rArr f(x) =(x cos x-sinx)/(x^2)`
`f(x)=g(x)/x^2,where g(x)=x cosx-sinx`
Now g(x) = x cos x-sin x
`rArr f(x)=-xsinx lt 0 "for" x in (0,pi//2)`
`rArr g(x) "is strictly decreasing in" [0,pi//2]`
`rArr g(x) lt g(0) " for all " x in (0,pi//2]`
`rArr g(x) lt 0 "for all " x in (0,pi//2]`
`f'(x) " is decreasing on x in" (0,pi//2]`
`rArr g(pi/2)lt f (x) le underset(xrarr0)limf(x)"for all " x in(0,pi//2]`
`rArr 2/pi lt (sinx)/(x)lt 1 "for all " x in (0,pi//2]`
Hence both the statements are true and statement-2 is a correct explanation of statement-1