Let `f` be a continuous function on `[a ,b]dot` Prove that there exists a number `x in [a , b]` such that `int_a^xf(t)dt=int_x^bf(t)dtdot`

Let f be continuous on [a , b],a >0,a n d differentiable on (a , b)dot Prove that there exists c in (a , b) such that (bf(a)-af(b))/(b-a)=f(c)-cf^(prime)(c)

Let f(x) be a continuous function AAx in R , except at x=0, such that int_0^a f(x)dx , ain R^+ exists. If g(x)=int_x^a(f(t))/t dt , prove that int_0^af(x)dx=int_0^ag(x)dx

STATEMENT 1: If f(x) is continuous on [a , b] , then there exists a point c in (a , b) such that int_a^bf(x)dx=f(c)(b-a) STATEMENT 2: For a < b , if ma n dM are, respectively, the smallest and greatest values of f(x)on[a , b] , then m(b-a)lt=int_a^bf(x)dxlt=(b-a)Mdot

Let f be a contiuous function on [a,b] . If F(x)=(int_(a)^(x)f(t)dt-int_(x)^(b)f(t)dt)(2x-(a+b)) , then prove that there exist some c epsilon(a,b) such that int_(a)^(c)f(t)dt-int_(c)^(b)f(t)dt=f(c)(a+b-2c)

Let f:[a,b]rarr[1,infty) be a continuous function and lt g: RrarrR be defined as g(x)={(0, "if x" le a),(int_a^x f(t)dt, "if a"lexleb),(int_a^b f(t)dt, "if x"gtb):} Then

Let, f:[a,b]to[1,oo) be a continuous function and let g:RR to RR be defined as g(x)={{:(0,"if "x lta","),(int_(a)^(x)f(t)dt,"if "alexleb),(int_(a)^(b)f(t)dt,"if "x gtb):} Then

Let f:[a,b]rarr[1,infty) be a continuous function and lt g: RrarrR be defined as g(x)={(0, "if x" lt a),(int_a^x f(t)dt, "if a"lexleb),(int_a^b f(t)dt, "if x"gtb):} Then a) g(x) is continuous but not differentiable at a b) g(x) is differentiable on R c) g(x) is continuous but nut differentiable at b d) g(x) is continuous and differentiable at either a or b but not both.

Let f: R rarr R be a continuous odd function, which vanishes exactly at one point and f(1)=1/2 . Suppose that F(x)=int_(-1)^xf(t)dt for all x in [-1,2] and G(x)=int_(-1)^x t|f(f(t))|dt for all x in [-1,2] . If lim_(x rarr 1)(F(x))/(G(x))=1/(14) , Then the value of f(1/2) is

Let f:RrarrR be a continuous odd function, which vanishes exactly at one point and f(1)=1/2. Suppose that F(x)=int_-1^xf(t)dt for all x in[-1,2] . If lim_(xto1)(F(x))/(G(x))=1/14 , then the value of f(1/2) is

If f(a+b-x)=f(x), then prove that int_a^b xf(x)dx=(a+b)/2int_a^bf(x)dxdot