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] .

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:StoS where S=(0,oo) be a twice differentiable function such that f(x+1) =f(x) . If g:StoR be defined as g(x)= log_(e)f(x) , then the value of |g''(5) -g''(1)| is equal to :

Let f be any continuously differentiable function on [a, b] and twice differentiable on (a, b) such that f(a)=f'(a)=0" and "f(b)=0 . Then-

Let f be any continuously differentiable function on [a,b] and twice differentiable on (a,b) such that f(a)=f'(a)=0 . Then

Let f be arry continuously differentiable function on [a,b] and twice differentiable on (a.b) such that f(a)=f(a)=0 and f(b)=0. Then

Let f:R rarr R be a twice differentiable function such that f(x+pi)=f(x) and f'(x)+f(x)>=0 for all x in R. show that f(x)>=0 for all x in R .