Real Analysis Answers

Determine whether the following functions are differentiable

i) 𝑓 (𝑥 )= |𝑥|;

ii) 𝑔(𝑥) = |𝑥| +| 𝑥 + 1 |

iii) h(x) = x^1/3

If I: = [0,4], calculate the norms of the following partitions:

a) P1: = (0,1,2,4)

b) P2: = (0,2,3,4)

c) P3: = (0,1,1.5,2,3.4,4)

d) P4: = (0,.5,2.5,3.5,4)

Test the following series for convergence

Σ from n=1to ♾️ for [√n⁴+9 - √n⁴-9]

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.