Define a continuous function on the closed interval [a,b]

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.

Define a function.

Define one to one function?

Find the absolute extrema of the function f(x)=3 cos x on the closed interval [0,2pi] .

Find the absolute extrema of the following functions on the given closed interval. f(x)=6x^(4/3)-3x^(1/3),[-1,1]

Find the absolute extrema of the following functions on the given closed interval. f(x)=2cosx+sin2x,[0,pi/2]

Find the absolute extrema of the following functions on the given closed interval. f(x)=x^(3)-12x+10,[1,2]

Find the absolute extrema of the following functions on the given closed interval. f(x)=3x^(4)-4x^(3),[-1,2]

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 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