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:(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, underset(x to x)lim(overset(1)underset(0)int sqrt(1(f(s))^(2))dxoverset(x)underset(0)int sqrt(1(f(s))^(2))ds)/(f(t)-f(x))=f(x) Then, the value of f((1)/(4)) belongs to

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

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

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

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] .

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] .

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] .

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

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]

If f(x) be a differentiable function for all positive numbers such that f(x.y)=f(x)+f(y) and f(e)=1 then lim_(x rarr0)(f(x+1))/(2x)