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

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)dx=int_x^bf(t)dtdot

If f(x) is differentiate in [a,b], then prove that there exists at least one c in (a,b)"such that"(a^(2)-b^(2))f'(c)=2c(f(a)-f(b)).

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.

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^(prime)(c))/(3c^2)=(f(b)-f(a))/(b^3-a^3)

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^(prime)(c))/(3c^2)=(f(b)-f(a))/(b^3-a^3)

Statement 1: If both functions f(t)a n dg(t) are continuous on the closed interval [a,b], differentiable on the open interval (a,b) and g^(prime)(t) is not zero on that open interval, then there exists some c in (a , b) such that (f^(prime)(c))/(g^(prime)(c))=(f(b)-f(a))/(g(b)-g(a)) Statement 2: If f(t)a n dg(t) are continuous and differentiable in [a, b], then there exists some c in (a,b) such that f^(prime)(c)=(f(b)-f(a))/(b-a)a n dg^(prime)(c)(g(b)-g(a))/(b-a) from Lagranges mean value theorem.

Prove that: int_a^b(f(x))/(f(x)+f(a+b-x))dx=(b-a)/2

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)

Let f be a continuous, differentiable, and bijective function. If the tangent to y=f(x) a tx=a is also the normal to y=f(x) a tx=b , then there exists at least one c in (a , b) such that (a)f^(prime)(c)=0 (b) f^(prime)(c)>0 (c) f^(prime)(c)<0 (d) none of these