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), prove that there is atleast one c in (a, b) , such that (f'(c))/(3c^(2))= (f(b)-f(a))/(b^(3)-a^(3)) .

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

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 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

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)

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

f is continuous in [a, b] and differentiable in (a, b) (where a>0 ) such that f(a)/a=f(b)/b. Prove that there exist x_0 in (a, b) such that f'(x_0 ) = f(x_0)/x_0