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Let f:(0,\ pi)->R be a twice differen...

Let `f:(0,\ pi)->R` be a twice differentiable function such that `(lim)_(t->x)(f(x)sint-f(x)sinx)/(t-x)=sin^2x` for all `x in (0,\ pi)` . If `f(pi/6)=-pi/(12)` , then which of the following statement(s) is (are) TRUE? `f(pi/4)=pi/(4sqrt(2))` (b) `f(x)<(x^4)/6-x^2` for all `x in (0,\ pi)` (c) There exists `alpha in (0,\ pi)` such that `f^(prime)(alpha)=0` (d) `f"(pi/2)+f(pi/2)=0`

A

`f((pi)/(4))=(pi)/(4sqrt(2))`

B

`f(x)lt(x^(4))/(6) "for all" x in(0,pi)`

C

There exists `alphain(0,pi) "such that" f'(alpha)=0`

D

`f''((pi)/(2))+f((pi)/(2))=0`

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