Real Analysis Answers

Let f: [-5,5]-->R be defined by f(x) =5[x] +x^2 , where [x] denotes the greatest integer function . Show that this function is integrable .Is this function also differentiable ? Justify your answer ?

Identify the intervals in which the function f on R defined by
f(x) = 4x^3-6x^2-72x+15 is both increasing and decreasing ?

Prove that between any two real roots of e^(3x) sin 5x =10?, there is at least one real root of e^(3x) cos5x+6=0 ?

Let f:R --> R be defined as f(x) ={ x^7sin(1/x) if x doesn't equal to 0 , 0 if x=0
Show that f"(0) exists and is equal to zero?

Examine whether the equation x^3- 13x +10 =0 has a real root in the interval ]-2,2[ ?

Show that the function f:]-1,1] [ --> R given by f(x) =x^3 is uniformly continuous and deduce that f is continuous at the point zero?

Check whether the following function is continuous or not:
f(x)={6x for 0<=x>5 , 7 for x= 5, 3x+15 for 5<x<=10}
Also determine the type of discontinuity, if it exits?

Show that the following series converges :
summation (infinity , n= 1) [ 1/(2n+3)(2n+5)]?

9.Let P be the set of positive real numbers and let Q be the set of negative real numbers. The union of P and Q, consist of all the real numbers except
a.3
b.zero
c.1
d.2

10.Let P(x) be any statement and let A be any set. Then there exists a set: {B={a|a\\in A,P} (a) is true}. The statement represents the
a.axiom of specification
b.Euclidean geometry axiom
c.None of the options
d.axiom of extension

5.Let S = {a, b, c, d} and T = {f, b, d, g}. Then the intersection of set S and T is
a.{b, d, f}
b.{b, d}
c.{b, d,c}
d.{b, d, g}

6.The … are those real numbers, which can be expressed as the ratio of two integers
a.natural numbers
b.irrational numbers
c.rational numbers
d.integers