Let `f:[0,4]rarrR` be a differentiable function then for some `alpha`, `beta` in `(0,2)` ` int_0^4 f(t)dt`

Let f:[0,4]rarr R be a differentiable function then for some alpha,beta in (0,2)int_(0)^(4)f(t)dt

Given a function f:[0,4]toR is differentiable ,then prove that for some alpha,beta epsilon(0,2), int_(0)^(4)f(t)dt=2alphaf(alpha^(2))+2betaf(beta^(2)) .

f:[0.4]rarr R is a differentiable function.Then prove that for some a,b in(0,4)f^(2)(4)-f^(2)(0)=8f'(a)*f(b)

Let F:[3,5] rarr R be a twice differentiable function on (3,5) such that F(x)= e^(-x) int_(3)^(x) (3t^(2)+2t+4F'(t)) dt . If F'(4)= (alpha e^(beta)-224)/((e^(beta)-4)^(2)) , then alpha+beta is equal to _____

Let f:(-1,1)rarrR be a differentiable function satisfying (f'(x))^4=16(f(x))^2 for all x in (-1,1) , f(0)=0. The number of such functions is

Let f(x) be a differentiable function in the interval (0,2) , then the value of int_0^2 f(x) dx is :

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

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

Let f:(0,oo)rarr(0,oo) be a differentiable function satisfying,x int_(x)^(x)(1-t)f(t)dt=int_(0)^(x)tf(t)dtx in R^(+) and f(1)=1 Determine f(x)