Total No. of Printed Pages—7
3 SEM TDC MTMH (CBCS) C 5
2 0 2 0
( Held in April–May, 2021 )
MATHEMATICS
( Core )
Paper : C–5
( Theory of Real Functions )
Full Marks : 80
Pass Marks : 32
Time : 3 hours
The figures in the margin indicate full marks
for the questions
1. (a) Define cluster point of a set A. 1
(b) Using the definition of limit to show that
lim ( )
x
x x
®
+ =
2
2 4 12
2
(c) If a set A Í R and f : A ® R has a limit
at C Î R, then prove that f is bounded
on some neighbourhood of C. 3
16-21/466 ( Turn Over )
( 2 )
(d) Evaluate the following limit (any one) : 3
(i) lim
x
x
®¥ x
-
+
5
3
, x > 0
(ii) lim
x
x x
® x x
+ - +
0 +
2
1 2 1 3
2
, x > 0
(e) Let I be an interval and let f : I ® R be
continuous on I, if a, b Î I and if k Î R
satisfies f (a) < k < f (b), then there exists
a point c Î I between a and b, prove that
f (c ) = k. 4
Or
State and prove preservation of intervals
theorem.
2. (a) Prove that the constant function
f (x ) = b is continuous on the set of real
number R. 1
(b) Write the type of discontinuity if
lim ( )
x c
f x
®
exists but not equal to f (c ). 1
(c) Define uniform continuity of a function. 2
16-21/466 ( Continued )
3 )
(d) Let A Í R, let f and g be two continuous
functions at x = c on A to R. Prove that
f + g is continuous at x = c. 3
3. (a) A function f : A ® R is continuous at
the point c Î A and if for every sequence
{ x } n
in A that converges to c. Prove that
the sequence { f (x )} n
converges to f (c ). 5
Or
State and prove location of roots
theorem.
(b) Let I be a closed bounded interval and
let f : I ® R be continuous on I. Then
prove that f is uniformly continuous
on I. 5
Or
Test the following function for
continuity at x = 0 :
f x
x
x x
x
( )
,
sin ,
=
=
¹
ì
í
î
0 0
0
1
4. (a) If a function f is differentiable at c, then
choose the correct answer : 1
(i)
f b f a
b a
f c
( ) ( )
( )
-
-
= ¢
16-21/466 ( Turn Over )
( 4 )
(ii) ¢ =
-
® -
f c
f x f c
x c x c
( ) lim
( ) ( )
, provided limit
exists
(iii) lim ( ) ( )
x c
f x f c
®
=
(iv) lim ( ) lim ( )
x c x c
f x f x
® + ® -
=
(b) If f : I ® R has a derivative at c Î I, then
prove that f is continuous at c. 2
(c) Let c be an interior point of the interval I
at which f : I ® R has relative
maximum. If the derivative of f at c
exists, then prove that f ¢(c) = 0. 3
(d) State and prove Caratheodory’s
theorem. 4
Or
Prove that if f : R ® R is an even
function and has a derivative at every
point, then f ¢ is an odd function and
vice versa.
16-21/466 ( Cont
5 )
5. (a) Find the derivative of
f (x ) = 5 - 2x + x
2
1
(b) Write the statement of Rolle’s theorem. 2
(c) If f (x ) and g (x ) are continuous on
I = [a, b], they are differentiable on (a, b)
and f ¢(x ) = g ¢(x ) for all x Î (a, b), then
there exists a constant k, prove that
f (x ) = g (x ) + k on I. 3
(d) State and prove Lagrange’s mean value
theorem. 5
Or
If f is differentiable on I = [a, b] and if k
is a number between f ¢(a) and f ¢(b),
then there is at least one point c in (a, b).
Prove that f ¢(c ) = k.
(e) Applying mean value theorem, prove
that -x £ sin x £ x, for x ³ 0. 4
Or
Verify Rolle’s theorem for the following
function :
f (x ) = x - x + x -
3 2 6 11 6, x Î [1,3]
16-21/466 ( Turn Over )
( 6 )
6. (a) Write the remainder after n terms of
Taylor’s theorem in Cauchy’s form. 1
(b) Deduce mean value theorem from
Cauchy’s mean value theorem. 2
(c) Verify Cauchy’s mean value theorem for
the functions f (x ) = x
2
, g (x ) = x
3
in the
interval [1, 2]. 4
(d) Let I Í R be an open interval, let
f : I ® R be differentiable on I and f ¢¢(a)
exists at a Î I. Show that
lim
( ) ( ) ( )
( )
h
f a h f a f a h
h
f a
®
+ - + -
= ¢¢
0 2
2
5
Or
State and prove Cauchy’s mean value
theorem.
7. (a) Write the Maclaurin’s series for the
expansion of f (x ) as a power series in x. 1
(b) Define convex function. 2
16-21/466 ( Continued )
If you download this question paper ,you can click the following word
