Let `f: [0,pi/2rarr[0,1]` be a differentiable function such that `f(0) = 0, f (pi/2)=1,` then

Let f:R rarr R be a differentiable function such that f(0)=0,f(((pi)/(2)))=3 and f'(0)=1 If g(x)=int_(x)^((pi)/(2))[f'(t)cos ect-cot t cos ectf(t)]dt for x in(0,(pi)/(2)] then lim_(x rarr0)g(x)=

Let f:R rarr R be a differentiable function such that f(0),f((pi)/(2))=3 and f'(0)=1. If g(x)=int_(x)^((pi)/(2))[f'(t)csc t-cot t csc tf(t)]dt for x in(0,(pi)/(2)] then lim_(x rarr0)g(x)=

Let f:R rarr R be a twice differential function such that f((pi)/(4))=0,f((5 pi)/(4))=0 and f(3)=4 then show that there exist a c in(0,2 pi) such that f'(c)+sin c-cos c<0

Let f:(0, oo)rarr(0, oo) be a differentiable function such that f(1)=e and lim_(trarrx)(t^(2)f^(2)(x)-x^(2)f^(2)(t))/(t-x)=0 . If f(x)=1 , then x is equal to :

Let f:R->R be a twice differential function such that f(pi/4)=0,f((5pi)/4)=0 and f(3)=4 , then show that there exist a cin(0,2pi) such that f^(primeprime)(c)+sinc-cosc<0

Let f:R rarr R be a differentiable and strictly decreasing function such that f(0)=1 and f(1)=0. For x in R, let F(x)=int_(0)^(x)(t-2)f(t)dt

Let f:(0,pi) rarr R be a twice differentiable function such that lim_(t rarr x) (f(x)sint-f(t)sinx)/(t-x) = sin^(2) x for all x in (0,pi) . If f((pi)/(6))=(-(pi)/(12)) then which of the following statement (s) is (are) TRUE?

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 .

Let f: (-5,5) rarr R be a differentiable function with f(4) = 1, f^(')(4) = 1, f(0) = -1 and f^(')(0) = 1 , If g(x) = f(2f^(2)(x)+2))^(2) , then - g^(')(0) equals