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(x)(b-a)` STATEMENT 2: For `a

If f is a continuous function on the interval [a,b] and there exists some c in(a,b) then prove that int_(a)^(b)f(x)dx=f(c)(b-a)

Let f be a continuous function on [a,b]. Prove that there exists a number x in[a,b] such that int_(a)^(x)f(t)dx=int_(x)^(b)f(t)dt

If f is continuous on [a,b] and differentiable in (a,b) then there exists c in(a,b) such that (f(b)-f(a))/((1)/(b)-(1)/(a)) is

Statement 1: If 27 a+9b+3c+d=0, then the equation f(x)=4a x^3+3b x^2+2c x+d=0 has at least one real root lying between (0,3)dot Statement 2: If f(x) is continuous in [a,b], derivable in (a , b) such that f(a)=f(b), then there exists at least one point c in (a , b) such that f^(prime)(c)=0.

int_(a + c)^(b+c) f(x)dx=

If f(x) is continuous in [a,b] and differentiable in (a,b), then prove that there exists at least one c in(a,b) such that (f'(c))/(3c^(2))=(f(b)-f(a))/(b^(3)-a^(3))

Statement-1 : If f is differentiable on an open interval (a,b) such that |f'(x)le M "for all" x in (a,b) , then |f(x)-f(y)|leM|x-y|"for all" in (a,b) Satement-2: If f(x) is a continuous function defined on [a,b] such that it is differentiable on (a,b) then exists c in (a,b) such that f'(c)=(f(b)-f(a))/(b-a)

int_(a)^(b)(f(x))/(f(x)+f(a+b-x))dx=

Let f be continuous on [a,b],a>0, and differentiable on (a,b). Prove that there exists c in(a,b) such that (bf(a,b))/(b-a)=f(c)-cf'(c)

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