Real Analysis Answers

Define a partition of an interval. Write any three different examples of

partitions of [0,1].

(ii). Let a real valued function 𝑓 be defined and bounded over [𝑎, 𝑏]. Prove that 𝑓 is

Riemann integrable over [𝑎, 𝑏] if for each 𝜖 > 0 there is a partition 𝑃 such that

𝑈(𝑃, 𝑓) − 𝐿(𝑃, 𝑓) < 𝜖

(iii). Show that 𝑓(𝑥) = 𝑥2

is Riemann integrable over [0,2].

Let f [: − 3,3 ] → R be defined by f (x)= 5x + x³ where [x] denotes the greatest integer ≤ x. Show that this function is integrable.

Show that there is no real number, k for which the equation, x⁴ - 3x² + k =0 has two distinct roots in the interval [2,3].

Using the principle of mathematical induction, prove that

(n⁵/5) + (n³/3) + (7n/15) is a natural number, ∀ n ∈ N .

Find lim x→0 (1-cos²x) / (x² sinx²)

Prove that the union of two closed sets is a closed set. Give an example to show that

union of an infinite number of closed sets need not be a closed set.

Examine the function f : R→R defined by

f(x) = { (1/6) (x+1)³ x≠ 0

f(x) = { 5/6 x=0

for continuity on R. If it is not continuous at any point of R, find the nature of

discontinuity there.

Show that the 2^n is not convergent.

The function f: R^2→R, defined by

f(x,y)= 1-x^2+y^2 has an extremum at (0,0)

True or false with full explanation

A small rectangular warehouse is to be constructed which is to have an area of 10000 square feet. The building is to be partitioning internally in to eight equal parts. The costs have been estimated based on exterior and interior walls dimensions. The costs are $200 per running foot of exterior wall and plus $100 per running foot of interior wall.