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 be a continuous function on [a , b]dot If F(x)=(int_a^xf(t)dt-int_x^bf(t)dt)(2x-(a+b)), then prove that there exist some c in (a , b) such that int_a^cf(t)dt-int_c^bf(t)dt=f(c)(a+b-2c)dot

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

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

Let f: RvecR be a continuous odd function, which vanishes exactly at one point and f(1)=1/2dot Suppose that F(x)=int_(-1)^xf(t)dtfora l lx in [-1,2]a n dG(x)=int_(-1)^x t|f(f(t))|dtfora l lx in [-1,2]dotIf(lim)_(xvec1)(F(x))/(G(x))=1/(14), Then the value of f(1/2) is

Show that int_a^b(|x|)/x dx=|b|-|a|dot

Let f be a function defined on the interval [0,2pi] such that int_(0)^(x)(f^(')(t)-sin2t)dt=int_(x)^(0)f(t)tantdt and f(0)=1 . Then the maximum value of f(x) is…………………..

f(x) is continuous function for all real values of x and satisfies int_0^xf(t)dt=int_x^1t^2f(t)dt+(x^(16))/8+(x^6)/3+adot Then the value of a is equal to: -1/(24) (b) (17)/(168) (c) 1/7 (d) -(167)/(840)

Let f(x) is continuous and positive for x in [a , b],g(x) is continuous for x in [a , b]a n dint_a^b|g(x)|dx >|int_a^bg(x)dx| STATEMENT 1 : The value of int_a^bf(x)g(x)dx can be zero. STATEMENT 2 : Equation g(x)=0 has at least one root for x in (a , b)dot

If" f, is a continuous function with int_0^x f(t) dt->oo as |x|->oo then show that every line y = mx intersects the curve y^2 + int_0^x f(t) dt = 2