If the function `f:[0,4]vecR` is differentiable, the show that for `a , b , in (0,4),f(4))^2-(f(0))^2=8f^(prime)(a)f(b)`

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)).

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)) .

If the function f : [0,8] to R is differentiable, then for 0 < alpha <1 < beta < 2 , int_0^8 f(t) dt is equal to (a) 3[alpha^3f(alpha^2)+beta^2f(beta^2)] (b) 3[alpha^3f(alpha)+beta^2f(beta)] (c) 3[alpha^3f(alpha^2)+beta^2f(beta^3)] (d) 3[alpha^2f(alpha^3)+beta^2f(beta^3)]

Let f:[2,7]vec[0,oo) be a continuous and differentiable function. Then show that (f(7)-f(2))((f(7))^2+(f(2))^2+f(2)f(7))/3=5f^2(c)f^(prime)(c), where c in [2,7]dot

If f:RR-> RR is a differentiable function such that f(x) > 2f(x) for all x in RR and f(0)=1, then

Let g(x) be the inverse of an invertible function f(x), which is differentiable for all real xdot Then g^('')(f(x)) equals. (a) -(f^('')(x))/((f^'(x))^3) (b) (f^(prime)(x)f^('')(x)-(f^(prime)(x))^3)/(f^(prime)(x)) (c) (f^(prime)(x)f^('')(x)-(f^(prime)(x))^2)/((f^(prime)(x))^2) (d) none of these

If f is a continuous function on [0,1], differentiable in (0, 1) such that f(1)=0, then there exists some c in (0,1) such that cf^(prime)(c)-f(c)=0 cf^(prime)(c)+cf(c)=0 f^(prime)(c)-cf(c)=0 cf^(prime)(c)+f(c)=0

For every function f (x) which is twice differentiable , these will be good approximation of int_(a)^(b)f(x)dx=((b-a)/(2)){f(a)+f(b)} , for more acutare results for cin(a,b),F( c) = (c-a)/(2)[f(a)-f( c)]+(b-c)/(2)[f(b)-f( c)] When c= (a+b)/(2) int_(a)^(b)f(x)dx=(b-a)/(4){f(a)+f (b)+2 f ( c) }dx If lim_(t toa) (int_(a)^(t)f(x)dx-((t-a))/(2){f(t)+f(a)})/((t-a)^(3))=0 , then degree of polynomial function f (x) atmost is

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] .