Real Analysis Answers

Test the following series for convergence.

Σ from n=1 to ♾️ for n.x^n-1 , x>0

Suppose that f:[0,2]→ R is continuous on [0,2] and differentiable on [0,2] and

that f(0) =0 , f(1) =1, f(2) =1.

(i) Show that there exists c↓1∈ (0,1)such that f'(c↓1) =1

(ii) Show that there exists c↓2 ∈ (1,2)such that f'(c↓2) =0.

(iii) Show that there exists c ∈ (0,2)such that f'(c) =1/3

Let f be a differentiable function on [α, β ] and x ∈[α, β ] .Show that, if

f ′(x) = 0 and , f ′′(x) >0 then f must have a local maximum at x.

Let f: [0, 1]→R be a function defined by f(x) = x^m (1-x)^n ,where . m, n∈N

Find the values of m and n such that the Rolle’s Theorem holds for the function

f .

Determine the local minimum and local maximum values of the function f defined by

f(x) = 3-5x³ +5x⁴ -x^5

Prove that a strictly decreasing function is always one-one.

Find the following limit.:

Lim x →0 for 1-cos x²/x².sin x²

Prove the limit of 𝑥𝑛 =1/2 [𝑥𝑛−1 + 𝑥𝑛]

let ck>0 for k=1,2,....,n.show that( c1+c2+....cn)/√n≤[ c1^2+c2^2+....+cn^2]^1/2≤c1+c2+...+cn

Find all real numbers x such that

1<x^2<4