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

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 f^(prime)(x) vanishes at least twice on [0,1] f^(prime)(1/2)=0 int_(-1/2)^(1/2)f(x+1/2)sinxdx=0 int_(-1/2)^(1/2)f(t)e^(sinpit)dt=int_(1/2)^1f(1-t)e^(sinpit)dt

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

Let f(x0 be a non-constant thrice differentiable function defined on (-oo,oo) such that f(x)=f(6-x)a n df^(prime)(0)=0=f^(prime)(x)^2=f(5)dot If n is the minimum number of roots of (f^(prime)(x)^2+f^(prime)(x)f^(x)=0 in the interval [0,6], then the value of n/2 is___

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 :

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

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)=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) =

If f:R->R is a twice differentiable function such that f''(x) > 0 for all x in R, and f(1/2)=1/2 and f(1)=1, then