Real Analysis Answers

show that the length of the curve y = log Sec x between the points x = 0 and x = pi/3 is log(2 + √3)

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

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).