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(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(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-

If y=f(x) is an odd differentiable function defined on (-oo,oo) such that f^(prime)(3)=-2,t h e n|f^(prime)(-3)| equals_________.

Let f: RvecR be a one-one onto differentiable function, such that f(2)=1a n df^(prime)(2)=3. The find the value of ((d/(dx)(f^(-1)(x))))_(x=1)

Let f: RvecR be a one-one onto differentiable function, such that f(2)=1a n df^(prime)(2)=3. The find the value of ((d/(dx)(f^(-1)(x))))_(x=1)

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 __________

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)=