Real Analysis Answers

Let X be a nonempty set and let f: X-> R have bounded ranges in r. if a element R. show that : sup {a + f(x): x element X} = a + sup {f(x): x element X} .Show that we also have inf {a+f(x) : x element X} = a+ inf {f(x): x element X}

Show that the length of the curve y=log sec x between the points x=0 and x=π/3 is log (2+√3).

Please find the maximum and minimum values of the function f(x)=(100+x^2)^(1/2)-x over non-negative x.

Let ε>0 and δ>0 and a Є R. Show that Vε(a)nVδ(a) and Vε(a)UVδ(a) are γ neighborhoods of a for appropriate values of γ.

let a1=a2=5 and an+1=an +6an-1 ,n>=2.show by induction that an=3n-(-2)n if n>=1.

prove that if a∫b f(x)dx exists as a proper integral ,then limx∫b f(t)dt=0.
x→b-

Consider all monic polynomials f(x)=x^2+bx+c, where b and c are real numbers. What is the minimum value of N, where

N=max x∈[−10,10]|f(x)|?

prove the following equality:
sup(f) = -inf(-f).

1) in the system of real numbers the axiom of existence of additive inverse states that, for all x element of R there exists y element of R such that x+y=y+x=0. prove that the additive inverse(y) corresponding to each real number x is unique. what can you say about the statement, there exists y element of R such that for all x element of R x+y=y+x=0?

2)if a and b are irrational numbers is, a to the power of b necessarily an irrational number? prove your claim

3)suppose A,B,C,D are four distinct points with position vectors a,b,c,d respectively, show that A,B,C,D lie on a plane if and only if there exists w,x,y,z element of R such that w+x+y+z=0 and aw+bx+cy+dz=0