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

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

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

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)

STATEMENT 1: If f(x) is continuous on [a,b] then there exists a point c in(a,b) such that int_(a)^(b)f(x)dx=f(x)(b-a) STATEMENT 2: For a

If f(x) and g(x) are continuous functions in [a,b] and are differentiable in (a,b) then prove that there exists at least one c in(a,b) for which.det[[f(a),f(b)g(a),g(b)]]=(b-a)det[[f(a),g'(c)]], where a

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