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 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 .

STATEMENT - 1 : Let f be a twice differentiable function such that f'(x) = g(x) and f''(x) = - f (x) . If h'(x) = [f(x)]^(2) + [g (x)]^(2) , h(1) = 8 and h (0) =2 Rightarrow h(2) =14 and STATEMENT - 2 : h''(x)=0

Let f be a twice differentiable function such that f''(x)=-f(x)" and "f'(x)=g(x)."If "h'(x)=[f(x)]^(2)+[g(x)]^(2), h(a)=8" and "h(0)=2," then "h(2)=

Let f be a twice differentiable function such that f''(x)=-f(x), and f'(x)=g(x),h(x)=[f(x)]^(2)+[g(x)]^(2) Find h(10) if h(5)=11

Let f be any continuous function on [0, 2] and twice differentiable on (0, 2). If f(0) = 0, f(1) = 1 and f(2) = 2, then :