Answer to Question #67006 in Real Analysis for koel

Question

5. (a) Define a compact set. Check whether the
set of integers is compact or not. 2
(b) If a + b + c = — 4 and 4a + 2b + c = 6, then
show that both the roots of the quadratic
equation axe + bx + c = 0 are real.
(c) Using Taylor's Theorem, prove that
, x2 x4
+ V XE R.
2! 4!
6. (a) State the second mean value theorem of
integrability. Verify it for the functions
f and g defined on [1, 2] by fix) = 3x and
g(x) = 5x.
(b) Test the following series for absolute and
conditional convergence : 5
00
(i) E E1)n
3n+1
n=1
E00 sin nx
n3
n=1
4
4
5
MT E-09 4
7. (a) Prove that there is no rational number
whose square is 6. 3
(b) Check whether the following functions are
continuous or not at x = 0. Also, find the
nature of discontinuity at that point, if it
exists. 4
- 2 x
(i) f(x) =
1
1
X
, x 0
, x=0
X
2 + -1 x<_0 ,
(ii) f(x) =
3
- (X3 -F ) , x>0 3
(c) Examine the function f given by
ME) = (x - 8)3 (x + 3), x E R
for extreme values. .