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

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

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 :

Let f(x) be a non-constant thrice differential function defined on (-oo,oo) such that f((x+13)/(2))=f((3-x)/(2)) and f'(0)=f'((1)/(2))=f'(3)=f'((9)/(2))=0 then the minimum number of zeros of h(x)=(f'(x))^(2)+f'(x)f''(x) in the interval [0,9] is 2k then k is equal to

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