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) 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) and f'((1)/(4))=0. Then (a) f'(x) vanishes at least twice on [0,1](b)f'((1)/(2))=0( c) int_(-(1)/(2))^((1)/(2))f(x+(1)/(2))sin xdx=0 (d) int_(-(1)/(2))^((1)/(2))f(t)e^(sin pi t)dt=int_((1)/(2))^(1)f(1-t)e^(sin pi t)dt

If int_(0)^(x)f(t)dt=e^(x)-ae^(2x)int_(0)^(1)f(t)e^(-t)dt , then

Let f(x) be a differentiable function such that f(x)=x^(2)+int_(0)^(x)e^(-t)f(x-t)dt then int_(0)^(1)f(x)dx=

If f(x)=int_(0)^(x){f(t)}^(-1)dt and int_(0)^(1){f(t)}^(-1)=sqrt(2)

If int_(0)^(1) f(t)dt=x^2+int_(0)^(1) t^2f(t)dt , then f'(1/2)is

if f(x) is a differential function such that f(x)=int_(0)^(x)(1+2xf(t))dt&f(1)=e , then Q. int_(0)^(1)f(x)dx=