Suppose `f` is a continuous map from `R` to `R` and `f(f(a))=a` for some `adot` Show that there is some `b` such that `f(b)=bdot`

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

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

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)

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 be continuous on [a, b] and assume the second derivative f" exists on (a, b). Suppose that the graph of f and the line segment joining the point (a,f(a)) and (b,f(b)) intersect at a point (x_0,f(x_0)) where a < x_0 < b. Show that there exists a point c in(a,b) such that f"(c)=0.

Tet f be continuous on [a,b] and differentiable on (a,b). If f(a)=a and f(b)=b, then

Suppose that f(x) is continuous in [0, 1] and f(0) = 0, f(1) = 0 . Prove f(c) = 1 - 2c^(2) for some c in (0, 1)