For any acute angled `triangleABC,(sinA)/(A)+(sinB)/(B)+(sinC)/(C )` can

Options

A. 1
B. 2
C. 3
D. 4

Answer

1,2

Explanation

Let `f(x)=(sinx)/(x)`
`rarr f(x)=(xcosx-sinx)/(x^(2))=(cosx(x-tanx))/(x^(2))`
`rarr f(X)=(-x^(2)sinx-2xcosx+2 sinx)/(x^(3))`
g(x)=`-x^(2)sinx-2xcosx+2 sinx`
`g(X)=-x^(2)cosxlt0 forall x in (0,pi//2)`
`rarr for xgt0` we have `g(X) lt g(0)i.e g(x)lt0`
`rarr f(X)lt0andf(x)lt0,x in (0,(pi)/(2))`
`rarr f(a+B+C)/(3)gtf(A)+f(B)+f(C )/(3)`
`rarr (sinA)/(A)+(sinB)/(B)+(sinC)/(C )le(9sqrt(3))/(2pi)=2.5`