Question

The fucntion f(x)`=(sin x)/(x)` is decreasing in the interval
A. (-lt/2, 0 )
B. `(0, pi//2)`
C. `(0, pi)`
D. none of these

Answer 1

Correct Answer - B::C
we have
` f(x) =(sin x)/(x)`
`rArr f(x)=g(x)/x^2` , where g(x)=x cos x - sin x .
Now g(x)=- sin x
Consider the interval `(-pi //2,0)`
In this inetrval we obseve that
`g(x) lt g( 0)`
`rArr` g(x) is decresing on `(- pi //2,0)`
`rArr g(x) gt 0 for all x in (-pi//2, 0)`
`therefore f(x)=(g(x)/(2) gt 0 for all x in (-pi //2 , 0)`
`rArr f(x) " in creasing on" (-pi //2 , 0)`
Consider now the interval `(0,pi//2)`
In this interval ,we have
`g(x) lt 0 "for all " x in (0 pi//2)`
`rAr g(x) "decresing on " (0,pi//2)`
`f(x) lt 0 for x in (0,pi//2)`