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`

Show that f(x) = sinx is continuous on R .

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)

If f is a function from R rarr R such that f(x) = x^2 , then show that f is not one-one .

Suppose that f: R to R is a continuous periodic function and T is the period of it . Let a in R . Then prove that for any positive integer n int_(0)^(a+nT) f(x)dx=n int_(a)^(a+T) f(x) dx

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

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)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