Let `g(x)=f(x)sinx ,w h e r ef(x)` is a twice differentiable function on `(-oo,oo)` such that `f(-pi)=1.` The value of `|g^''(-pi)|` equals __________

Let g(x)=f(x)sinx ,w h e r ef(x) is a twice differentiable function on (-oo,oo) such that f'(-pi)=1. The value of g''(-pi) equals __________

Let g(x) = ln f(x) where f(x) is a twice differentiable positive function on (0, oo) such that f(x+1) = x f(x) . Then for N = 1,2,3 g''(N+1/2)- g''(1/2) =

Let f(x) be a non-constant twice differentiable function defined on (-oo,oo) such that f(x)=f(1-x)a n df^(prime)(1/4)=0. Then

If y=f(x) is an odd differentiable function defined on (-oo,oo) such THAT f'(3)=-2 then f'(-3) equal to-

Let f(x) be a non-constant twice differentiable function defined on (oo, oo) such that f(x) = f(1-x) and f"(1/4) = 0 . Then

Leg g(x)=ln(f(x)), whre f(x) is a twice differentiable positive function on (0,oo) such that f(x+1)=xf(x)dot Then, for N=1,2,3, g^(N+1/2)-g^(1/2)=

Let f (x) be a twice differentiable function defined on (-oo,oo) such that f (x) =f (2-x)and f '((1)/(2 )) =f' ((1)/(4))=0. Then int _(-1) ^(1) f'(1+ x ) x ^(2) e ^(x ^(2))dx is equal to :

Prove that the curves y=f(x),[f(x)>0],a n dy=f(x)sinx ,w h e r ef(x) is differentiable function, have common tangents at common points.

If f(x) is a twice differentiable function such that f'' (x) =-f(x),f'(x)=g(x),h(x)=f^2(x)+g^2(x) and h(10)=10 , then h (5) is equal to

Let f(x)=cot^-1g(x)] where g(x) is an increasing function on the interval (0,pi) Then f(x) is