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)`

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)a n dg(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. |f(a)f(b)g(a)g(b)|=(b-a)|f(a)f^(prime)(c)g(a)g^(prime)(c)|,w h e r ea

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)).

If f is a continuous function on [0,1], differentiable in (0, 1) such that f(1)=0, then there exists some c in (0,1) such that cf^(prime)(c)-f(c)=0 cf^(prime)(c)+cf(c)=0 f^(prime)(c)-cf(c)=0 cf^(prime)(c)+f(c)=0

Statement 1: If both functions f(t)a n dg(t) are continuous on the closed interval [1,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 continuou 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 Lagranes mean value theorem.

Let f and g be function continuous in [a,b] and differentiable on [a,b] .If f(a)=f(b)=0 then show that there is a point c in (a,b) such that g'(c) f(c)+f'(c)=0 .

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 a , b , c be three real numbers such that a

Leg f be continuous on the interval [0,1] to R such that f(0)=f(1)dot Prove that there exists a point c in [(0,1)/2] such that f(c)=f(c+1/2)dot

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.