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 be a continuous function defined onto [0,1] with range [0,1]. Show that there is some c in [0,1] such that f(c)=1-c

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 .

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.

Let f,\ g and h be functions from R to R . Show that (f+g)oh=foh+goh

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

Suppose f is continuous on [a,b] & f(a)=f(b)=0 & int_(a)^(b)f^(2)(x)dx=1 then the minimum value of int_(a)^(0)(f^(')(x))^(2)dx int_(a)^(b) x^(2)f^(2)(x)dx is k , then 8k is

Let f(x) be a continuous function such that f(0) = 1 and f(x)=f(x/7)=x/7 AA x in R, then f(42) is

If a continuous function f defined on the real line R assume positive and negative values in R, then the equation f(x)=0 has a root in R. For example, if it is known that a continuous function f on R is positive at some point and its minimum value is negative, then the equation f(x)=0 has a root in R. Consider f(x)= ke^(x)-x , for all real x where k is a real constant. The positive value of k for which ke^(x)-x=0 has only one root is

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

Let f: R->R be a continuous function and f(x)=f(2x) is true AAx in Rdot If f(1)=3, then the value of int_(-1)^1f(f(x))dx is equal to (a)6 (b) 0 (c) 3f(3) (d) 2f(0)