Let `f : [0, 1] rarr [0, 1] `be a continuous function such that `f (f (x))=1 for all x in[0,1]`then:

Let f:[0,1]rarr[0,1] be a continuous function such that f(f(x))=1f or allx in[0,1] then

Let f:[0,1]rarr[0,1] be a continuous function. Then prove that f(x)=x for at least one 0<=x<=1

Let f(x) be a continuous function such that f(a-x)+f(x)=0 for all x in [0,a] . Then int_0^a dx/(1+e^(f(x)))= (A) a (B) a/2 (C) 1/2f(a) (D) none of these

Let f(x) be a continuous function such that f(a-x)+f(x)=0 for all x in [0,a] . Then, the value of the integral int_(0)^(a) (1)/(1+e^(f(x)))dx is equal to

Let f(x) be the continuous function such that f(x)= (1-e^(x))/(x) for x!=0 then

Let f:(0,1) in (0,1) be a differenttiable function such that f(x)ne 0 for all x in (0,1) and f((1)/(2))=(sqrt(3))/(2) . Suppose for all x, lim_(x to x)(int_(0)^(1) sqrt(1(f(s))^(2))dxint_(0)^(x) sqrt(1(f(s))^(2))ds)/(f(t)-f(x))=f(x) Then, the value of f((1)/(4)) belongs to

Let f defined on [0,1] be twice differentiable such that |f(x)|<=1 for x in[0,1], if f(0)=f(1) then show that |f'(x)<1 for all x in[0,1]