Let f:(0,pi)rarrR be a twice differentiable fucntion such that `lim_(trarrx) (f(x)sint-f(t)sinx)/(t-x) = sin^(2) x ` for all x `in (0,pi)`
If `f((pi)/(6))=(-(pi)/(12))` then which of the following statement (s) is (are) TRUE?

Options

A. `f(x(pi)/(4))=(pi)/(4sqrt(2))`
B. `f(x)lt(x^(4))/(6)-x^(2)for all x in(0,pi)`
C. There exist `alpha in (0,pi)` such that `f'(alpha)=0`
D. `f''((pi)/(2))+f((pi)/(2))`=0

Answer

2,3,4

Explanation

`underset(t rarr x)limf(x)sint-(f(t)sinx)/(t-x)=sin^(2)x`
`rarr underset(t rarr x)lim (fx)cosx-f(x)(sinx)/(sin^(2)x)=1`
`rarr (f(x))/(sinx)=-x+c`
since `f(xpi)/(6)=-(pi)/(12)c=0`
`rarr f(X)=-x sin x`
`f(pi)/(4)=(pi)/(4sqrt(2))`
Let `g(x)=-xcosx -sinx+2x-(2x^(3))/(3)`
`rarr g(X)=-2cosx +sinx+2x-(2x^(3))/(3)`
`rarr g(x)=3 sin x +x cos x -4x`
`=3(sinx x-x)+x(cosx-1)`
`rarr g(X)lt0`
Hence g(X) is decreasing
So for `xgt0g(x) ltg(0)`
hence g(X) is decreasing
So for `xgt0g(X)ltg(0)`
Hence g(X) is decreasing
so for `xgt0 g(X) ltg(0)`
`rarr g(X)lt0`
`hence f(X)lt(x^(4))/(6)-x^(2)forall x in (0,pi)`
f(x) is ocntinous and differentable and f(X) =`f(pi)=0`
using rolle theorem `f(alpha)=0 for some alpha in (0,oi)`