Continuity and Differentiability
A)NCERT Problems:
1)
2)
3)
4)
5)
6)
7)
|𝑥| + 3, 𝑖𝑓 𝑥 ≤ −3
Find all points of discontinuity of 𝑓 is defined by 𝑓(𝑥) = {−2𝑥, 𝑖𝑓 − 3 < 𝑥 < 3
6𝑥 + 2, 𝑖𝑓 𝑥 ≥ 3
Find the relationship between 𝑎 and 𝑏 so that the function 𝑓 defined by
𝑓(𝑥) =
𝑎𝑥 + 1, 𝑖𝑓 𝑥 ≤ 3
{
is continuous at 𝑥 = 3
𝑏𝑥 + 3, 𝑖𝑓 𝑥 > 3
𝜇(𝑥 2 − 2𝑥), 𝑖𝑓 𝑥 ≤ 0
For what value of 𝜇 is the function defined by 𝑓(𝑥) = {
continuous at 𝑥 = 0 ?
4𝑥 + 1, 𝑖𝑓 𝑥 > 0
What about continuity at 𝑥 = 1?
Show that the function defined by 𝑔(𝑥) = 𝑥 − [𝑥] is discontinuous at all integral points. Here [𝑥]
denotes the greatest integer less than or equal to 𝑥
sin 𝑥
, 𝑖𝑓 𝑥 < 0
Find all points of discontinuity of 𝑓, where 𝑓(𝑥) = { 𝑥
𝑥 + 1, 𝑖𝑓 𝑥 ≥ 0
1
2
𝑥 𝑠𝑖𝑛 𝑥 , 𝑖𝑓 𝑥 ≠ 0
Determine if 𝑓 defined by 𝑓(𝑥) = {
is a continuous function?
0
,𝑥 = 0
𝑘 cos 𝑥
𝜋
, 𝑖𝑓 𝑥 ≠ 2
𝜋
𝜋−2 𝑥
Find the values of 𝑘 so that the function 𝑓(𝑥) = {
is continuous at 𝑥 = 2
𝜋
3,
𝑖𝑓 𝑥 = 2
5,
𝑖𝑓 𝑥 ≤ 2
8) Find the values of 𝑎 and 𝑏 such that the function defined by 𝑓(𝑥) = {𝑎𝑥 + 𝑏, 𝑖𝑓 2 < 𝑥 < 10 is a
21,
𝑖𝑓 𝑥 ≥ 10
continuous function.
9) Show that the function defined by 𝑓(𝑥) = 𝑐𝑜𝑠(𝑥 2 ) is a continuous function.
10) Prove that the function 𝑓 given by 𝑓(𝑥) = |𝑥 − 1|, 𝑥 ∈ 𝑅 is not differentiable at 𝑥 = 1
11) Prove that the greatest integer function defined by 𝑓(𝑥) = [𝑥], 0 < 𝑥 < 3 is not differentiable at
𝑥 = 1 and 𝑥 = 2
(𝑥−3)(𝑥 2 +4)
12) Differentiate √
𝑑𝑦
3𝑥 2 +4𝑥+5
𝑦
𝑥
with respect to 𝑥
13) Find 𝑑𝑥 , if 𝑦 𝑥 +𝑥 + 𝑥 = 𝑎𝑏
14) Differentiate (log 𝑥)𝑥 + 𝑥 log 𝑥 with respect to 𝑥
15) Differentiate (sin 𝑥)𝑥 + sin−1 √𝑥 with respect to 𝑥
1
16) Differentiate (𝑥 cos 𝑥)𝑥 + (𝑥 sin 𝑥)𝑥 with respect to 𝑥
17) Differentiate 𝑥 sin 𝑥 + (sin 𝑥)cos 𝑥 with respect to 𝑥
𝑑𝑦
18) Find 𝑑𝑥 of (cos 𝑥)𝑦 = (cos 𝑦)𝑥
𝑑𝑦
19) Find 𝑑𝑥 of 𝑥𝑦 = 𝑒 (𝑥−𝑦)
20) Find
21) Find
𝑑𝑦
𝑑𝑥
𝑑𝑦
𝑑𝑥
and
and
𝑑2 𝑦
𝑑𝑥 2
𝑑2 𝑦
𝑑𝑥 2
𝑡
, 𝑥 = 𝑎 (cos 𝑡 + log 𝑡𝑎𝑛 2) and 𝑦 = 𝑎 sin 𝑡
, 𝑥 = 𝑎(cos 𝜃 + 𝜃 sin 𝜃) , 𝑦 = 𝑎(sin 𝜃 − 𝜃 cos 𝜃)
𝑑𝑦
−1
−1
22) If 𝑥 = √𝑎sin 𝑡 , 𝑦 = √𝑎 cos 𝑡 , show that 𝑑𝑥 = −
𝑦
𝑥
23) If 𝑦 = cos −1 𝑥. Find
𝑑2 𝑦
𝑑𝑥 2
in terms of 𝑦
24) If 𝑦 = 3𝑒 2𝑥 + 2𝑒 2𝑥 , prove that
𝑑2 𝑦
𝑑𝑦
- 5 𝑑𝑥 + 6𝑦 = 0
𝑑𝑥 2
𝑑2 𝑦
𝑑𝑦
25) If 𝑦 = sin−1 𝑥, show that (1 − 𝑥 2 ) 𝑑𝑥 2 − 𝑥 𝑑𝑥 = 0
26) If 𝑦 = 3 𝑐𝑜𝑠(𝑙𝑜𝑔𝑥) + 4 𝑠𝑖𝑛(𝑙𝑜𝑔𝑥), show that 𝑥 2 𝑦2 + 𝑥𝑦1 +𝑦 = 0
𝑑2 𝑦
𝑑𝑦 2
27) If 𝑒 𝑦 (𝑥 + 1) = 1, show that 𝑑𝑥 2 = (𝑑𝑥 )
28) If 𝑦 = (tan−1 𝑥)2 , show that (𝑥 2 + 1)2 𝑦2 + 2𝑥(𝑥 2 + 1)𝑦1 = 2
29) State Rolle’s theorem. Write the geometrical interpretation of Rolle’s theorem.
30) Verify Rolle’s theorem for the function 𝑓(𝑥) = 𝑥 2 + 2𝑥 − 8, 𝑥 ∈ [ −4, 2]
31) State Mean value theorem. Write the geometrical interpretation of Mean value theorem.
32) Verify Mean value theorem, if 𝑓(𝑥) = 𝑥 2 − 4𝑥 − 3 in the interval [𝑎, 𝑏], where 𝑎 = 1 and 𝑏 = 3.
33) If 𝑓: [ −5, 5] → 𝑅 is a differentiable function and 𝑓 ′ (𝑥) does not vanish anywhere , then prove that
𝑓(−5) ≠ 𝑓(5)
2𝑥+1
34) Differentiate 𝑙𝑜𝑔7 (log 𝑥) and sin−1 (1+ 4𝑥 ) with respect to 𝑥
1 𝑎
1
𝑑𝑦
35) For a positive constant 𝑎 find 𝑑𝑥 , where 𝑦 = 𝑎𝑡+ 𝑡 and 𝑥 = (𝑡 + 𝑡 )
36) Differentiate cot −1 [
𝑑𝑦
√1+sin 𝑥+√1−sin 𝑥
𝜋
√1+sin
2
] ,0 < 𝑥 <
𝑥 −√1−sin 𝑥
with respect to 𝑥
37) Find 𝑑𝑥 , if 𝑦 = sin−1 𝑥 + sin−1 √1 − 𝑥 2 , −1 ≤ 𝑥 ≤ 1
𝑑𝑦
1
38) If 𝑥√1 + 𝑦 + 𝑦√1 + 𝑥 = 0, for – 1< 𝑥 < 1, prove that 𝑑𝑥 = − (𝑥+1)2
39) If (𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑐 2, for some 𝑐 > 0, prove that
3
𝑑𝑦 2 2
[1+( ) ]
𝑑𝑥
𝑑2 𝑦
𝑑𝑥2
is a constant independent of 𝑎
and 𝑏
𝑐𝑜𝑠 2 (𝑎+𝑦)
𝑑𝑦
40) If cos 𝑦 = 𝑥 cos(𝑎 + 𝑦) , with cos 𝑎 ≠ ±1, prove that =
𝑑𝑥
41) If 𝑓(𝑥) = |𝑥|3, show that 𝑓 ′′ (𝑥) exists for all real 𝑥 and find it.
42) If 𝑦 = 𝑒 𝑎 cos
−1 𝑥
, −≤ 𝑥 ≤ 1, show that (1 − 𝑥 2 )2
𝑑2 𝑦
𝑑𝑥 2
-𝑥
𝑑𝑦
𝑑𝑥
sin 𝑎
− 𝑎2 𝑦 = 0
a) extra problems:
sin 3𝑥+𝑎 sin 2𝑥+𝑏𝑐𝑜𝑠𝑥
1)
If f(x) =
2)
A function f is defined as f(x) =
𝑥3
is continuous at x = 0, find the value of a and b
𝑥 2 − 4𝑥+3
𝑥 2 + 2𝑥−3
=3)
Let f(x) =
𝑥 3 + 𝑥 2 −16𝑥+20
( 𝑥−2)2
1
2
for x ≠ 1
for x = 1. Show that f(x) is differentiable at x = 1 and find its value
if x ≠ 2
= k, if x = 2.
If f(x) is continuous for all x, then find the value of k.
4)
Let f(x) be a function of x defined as f(x) =
=
Discuss the continuity of function at x = 1
5)
𝑥2 − 1
𝑥 2 − |𝑥|− 1
1
2
,x≠1
, x = 1.
Determine the values of a, b, c for which the function f(x) =
sin( 𝑎+1)𝑥+𝑠𝑖𝑛𝑥
𝑥
, for x < 0
= c , for x = 0
1
=
6)
f(x) =
, for x > 0
3
𝑏𝑥 2
1−𝑐𝑜𝑠4𝑥
,
𝑥2
= a,
=
1
( 𝑥+𝑏𝑥 2 )2 − 𝑥 2
x<0
x= 0
√𝑥
√16+ √𝑥 − 4
, x >0
is continuous at x = 0
𝑥 + 𝑎√2 𝑠𝑖𝑛𝑥,
7)
0≤𝑥<
𝜋
4
Find the value of a and b so that the function 𝑓(𝑥) = 2𝑥 cot 𝑥 + 𝑏,
≤ 𝑥≤
{ 𝑎 cos 𝑥 − 𝑏 sin 𝑥
𝜋
,2
𝜋
2
𝜋
4
is continuous at 0 ≤
<𝑥 ≤ 𝜋
x≤𝜋
𝑎+3 cos 𝑥
8)
Let 𝑓(𝑥) = {
𝑏 𝑡𝑎𝑛 (
𝑥2
𝜋
, 𝑥<0
), 𝑥 ≥ 0
where [ ] represents the greatest integer function. If f(x) is
[ 𝑥+3]
1
continuous at x = 0, then prove that a = -3 and b = - √3
2
9)
𝑎𝑥 2 − 𝑏, |𝑥| < 1
1
𝑓(𝑥) = {
. The above function is continuous and differentiable , then prove that a
− , |𝑥| ≥ 1
|𝑥|
1
3
2
2
= ,b=
10)
Discuss the continuity and differentiability of the function f(x) =
𝑥
1+ |𝑥|
=
11)
12)
3 𝑐𝑜𝑠2 𝑥
, 𝑖𝑓 𝑥 <
𝑎,
Let 𝑓(𝑥) =
𝑏 ( 1−sin 𝑥)
{
14)
(𝜋−2𝑥)2
𝑖𝑓 𝑥 =
𝜋
2
𝜋
, 𝑖𝑓 𝑥 >
Prove that function 𝑓(𝑥) = {
𝜋
. If f(x) be a continuous at x = 2 , find a and b.
2
𝜋
2
𝑥
|𝑥|+ 2𝑥 2
, 𝑖𝑓 𝑥 ≠ 0
𝑘,
is discontinuous at x = 0, regardless of the value of k.
𝑖𝑓 𝑥 = 0
𝑘 cos 𝑥
, 𝑖𝑓
𝜋−2𝑥
15)
Find the value of k, such that function ‘f’ defined by 𝑓(𝑥) = {
16)
Find the set of all points where the function 𝑓 (𝑥) = {
(cos 𝑥 − sin 𝑥)
17)
, |𝑥| ≤ 1
2 + √( 1 − 𝑥 2 ), |𝑥| ≤ 1
Discuss the continuity and differentiability of the function 𝑓(𝑥) = {
2
2𝑒 ( 1−𝑥) , |𝑥| > 1
𝑎𝑥 2 + 𝑏, 𝑖𝑓 𝑥 > 2
Determine the constants a and b, such that the function 𝑓(𝑥) = { 2,
𝑖𝑓 𝑥 = 2 is continuous
2𝑎𝑥 − 𝑏, 𝑖𝑓 𝑥 < 2
1− 𝑠𝑖𝑛3 𝑥
13)
, |𝑥| ≥ 1
𝑥
1− |𝑥|
3,
is continuous at x =
1+ 𝑒 𝑥
𝑐𝑜𝑠𝑒𝑐 𝑥
𝜋
, 𝑖𝑓 − < 𝑥 < 0
2
𝑖𝑓 𝑥 = 0
1
2
3
𝑒 𝑥+ 𝑒 𝑥+ 𝑒 𝑥
{
𝑖𝑓 𝑥 =
𝜋
2
𝜋
2
2
3
𝑎 𝑒𝑥 + 𝑏 𝑒𝑥
𝜋
2
0 , 𝑖𝑓 𝑥 = 0
𝑥
is differentiable.
1 , 𝑖𝑓 𝑥 ≠ 0
𝑎,
Let f(x) be defined as follows 𝑓(𝑥) =
𝑥 ≠
,
𝑖𝑓
𝑜<𝑥<
𝜋
2
. If f(x) is
continuous at x = 0, then find the value of and b
.
1
𝑥 (3𝑒 𝑥 +4 )
18)
, 𝑖𝑓 𝑥 ≠ 0
1
Discuss the continuity and differentiability of the function 𝑓 (𝑥) = { ( 2− 𝑒 𝑥 )
0 ,
19)
1
−1
𝑥 (𝑒 𝑥 −𝑒 𝑥 )
1
−1
( 𝑒𝑥 + 𝑒 𝑥 )
Let 𝑓 (𝑥) = {
0 ,
, 𝑖𝑓 𝑥 ≠ 0
𝑖𝑓 𝑥 = 0
. prove that f(x) is not differentiable at x = 0
𝑖𝑓 𝑥 = 0
y = (sinx) x + ( cosx ) tanx
20)
Differentiate w.r.t. x-
21)
Differentiate w.r.t. x-
22)
If xy = e x-y, prove that
y = (logx) x + x logx
𝑑𝑦
𝑑𝑥
=
log 𝑥
(1+log 𝑥)2
1
23)
at x = 0
Differentiate w.r.t. x - y = tan
−1
1
𝑥 3+ 𝑎3
(
1
1
)
1− 𝑎3 𝑥 3
cos 𝑥−sin 𝑥
24)
Differentiate w.r.t. x- y = tan−1 (cos 𝑥+ 𝑠𝑖𝑛𝑥)
25)
If y = √𝑥 + √𝑥 + √𝑥 + ⋯ … … … , prove that
26)
Differentiate y = 𝑥 𝑥 , w.r.t. x
27)
If y √1 − 𝑥 2 + x √1 − 𝑦 2 = 1 , prove that
𝑑𝑦
𝑑𝑥
𝑥
𝑥2
𝑑𝑥
1− 𝑥 2
𝑑𝑦
𝑥𝑛
𝑑𝑥
𝑛!
29)
If √1 − 𝑥 2 + √1 − 𝑦 2 = a( x- y) , prove that
30)
If sin−1(𝑥 2 √1 − 𝑥 2 + 𝑥 √1 − 𝑥 4 ) , then prove that
𝑛!
, show that
1− 𝑦 2
If y = 1 + x +
3!
+ ……… +
𝑥𝑛
=-√
2𝑦−1
28)
2!
+
𝑥3
𝑑𝑦
1
=
𝑑𝑦
𝑑𝑥
Differentiate w.r.t.x : tan−1
32)
If √1 − x 6 + √1 − y 6 = a3 ( x3 – y3), show that
33)
If x =
34)
√cos
cos3 θ
,y=
2t
1− 𝑦 2
1− 𝑥 2
𝑑𝑦
𝑑𝑥
=√
2𝑥
1−𝑥4
+√
1
1− 𝑥2
√1+ x2 − √1−x2
, find
√cos 2t
=0
√1+ x2 +√1−x2
31)
sin3 θ
=√
-y +
dy
dx
at t =
x2
dy
1− y6
= √ 1−x6
dx y2
𝜋
6
a−x
dy
a−x
If y = √( a − x)(x − b) - ( a – b) tan−1 √x−b , then prove that dx = √x−b
1
35)
If y = log [
36)
If y = 1 +
{
c1
( c1 − x)
√1+x+ √1−x 2
dy
√1+x− √1−x
c1
x c2
dx
] , then
x− c1
+
37) If y = 2 tan
x
2n
(
x
x
2
22
=
c2
x− c1 )( x− c2 )
c3
( c2 – x)
−1 x √2
If cos cos
tan
+(
1
2n
+
1− x2
x
cos
cot
x
2n
+
1
2x √1− x2
x2 c3
x− c1 )( x− c2 )( x− c3 )
c4
1−x √2+ x2
x
…………… cos
– cot x
+(
x3 c4
x− c1 )( x− c2 )( x− c3 )(x− c4 )
, show that
dy
dx
=
y
x
}
( c3 − x)
(c4 − x)
1+x √2+ x2
) + log
23
+(
=-
2n
, then
=
dy
dx
sin x
x
=
2n sin( n )
2
4√2
1+ x4
, prove that that
1
2
tan
x
2
+
1
22
tan
x
22
+ ………. +
1
2n
x n
y
38) If cos −1 (b) = log ( ) , prove that x 2 𝑦2 + x y1 n2 y = 0
n
39) If y = (sin−1 x)2 + ( cos−1 𝑥)2 , then prove that ( 1- x2 )𝑦2 - x y1 = 4
40) If y = tan−1 (
a1 x − α
a2 − a1
a1 α+x
1+ a1 a2
) + tan−1 (
tan−1(𝑎𝑛 ) , then find
𝑠𝑖𝑛𝑥
41) If y =
dy
) + tan−1 (
a3 − a2
1+ a2 a3
) + ……………. + tan−1 (
𝑎𝑛 − 𝑎𝑛−1
1+ 𝑎𝑛−1 𝑎𝑛
)-
dx
cos 𝑥
1+
sin 𝑥
1+ 1+cos 𝑥…………∞
, then prove that
1
dy
dx
=
( 1+𝑦) cos 𝑥+sin 𝑥
1+2𝑦+cos 𝑥−sin 𝑥
1
1
42) If y = tan−1 ( 𝑥 2 + 𝑥+1) + tan−1 (𝑥 2 + 3𝑥 + 3) + tan−1 (𝑥 2 + 5𝑥 +7) + …………………. To n terms, show that
dy
dx
=
1
(𝑥+𝑛)2 + 1
-
1
𝑥2+ 1
( 1 + x)a
43) If f( x) = |
1
(1 + 2x)b
(1 + 2x)b
( 1 + x)a
1
1
(1 + 2x)b | , then find i) constant term ii) coefficient of x
( 1 + x)a
sin( 𝑎+1)𝑥+𝑠𝑖𝑛𝑥
44)
Determine the values of a, b, c for which the function f(x) =
𝑥
= c , for x = 0
1
=
, for x < 0
1
( 𝑥+𝑏𝑥 2 )2 − 𝑥 2
3
, for x > 0
𝑏𝑥 2
𝑎𝑥 2 + 𝑏, 𝑖𝑓 𝑥 > 2
Determine the constants a and b, such that the function 𝑓(𝑥) = { 2,
𝑖𝑓 𝑥 = 2 is continuous
2𝑎𝑥 − 𝑏, 𝑖𝑓 𝑥 < 2
45)
46)
47)
If y √1 − 𝑥 2 + x √1 − 𝑦 2 = 1 , prove that
1
𝑑𝑦
𝑑𝑥
1
1− 𝑦 2
= - √1− 𝑥2
1
If y = tan−1 ( 𝑥 2 + 𝑥+1) + tan−1 (𝑥 2 + 3𝑥 + 3) + tan−1 (𝑥 2 + 5𝑥 +7) + …………………. To n terms, show that
1
1
(𝑥+𝑛)2 + 1 𝑥 2 + 1
b) extra problems( Advanced)
1) Let 𝑓(𝑥) =
𝑥
𝑎
𝑥
𝑏
𝑙𝑜𝑔(1+ )−𝑙𝑜𝑔(1− )
𝑥
, 𝑥 ≠ 0. Find the value of f at x = 0 so that f becomes continuous at x = 0
2) If y = √( a − x)(x − b) - ( a – b) tan−1 √
3) If x =
sin3 θ
√cos 2t
,y=
cos3 θ
,
√cos 2t
dy
a−x
x−b
, then prove that
dy
dx
=√
a−x
x−b
𝜋
find dx at t = 6
4) Find the values of a and b such that the function f(x) is defined by
𝜋
𝑥 + 𝑎√2𝑠𝑖𝑛𝑥, 0 ≤ 𝑥 < 4
𝜋
2𝑥 𝑐𝑜𝑡𝑥 + 𝑏, 4 ≤ 𝑥 ≤
{𝑎 𝑐𝑜𝑠2𝑥 − 𝑏
𝜋
𝑠𝑖𝑛𝑥, 2
𝜋
2
<𝑥≤𝜋
is continuous for all values of x and 0≤ 𝑥 ≤ 𝜋
𝑓(𝑥) =
dy
dx
=
(4𝑥 −1)3
5)
𝑥
𝑥2
𝑠𝑖𝑛( )𝑙𝑜𝑔(1+ )
𝑎
3
,𝑥 ≠ 0
Find the value of a for which the function 𝑓(𝑥) = {
may be continuous at x = 0
3
12(𝑙𝑜𝑔4) , 𝑥 = 0
6)
If y = √𝑥 + √𝑥 + √𝑥 + ⋯ … … … , prove that
7)
Differentiate tan−1 (
8)
If 𝑦 = (cos 𝑥)(cos 𝑥)
9)
If 𝑦 = tan−1 (
10)
If 𝑦 =
11)
If 𝑦 =
(cos 𝑥)…∞
Differentiate
13)
If 𝑦 =
to tan−1 𝑥 , when 𝑥 ≠ 0
𝑦 2 tan 𝑥
𝑑𝑦
, show that 𝑑𝑥= 𝑦𝑙𝑜𝑔(cos 𝑥)−1
𝑥+
𝑥+⋯.∞
√1+𝑥 2 +√1−𝑥 2
w.r.t. √1 − 𝑥 4
√1+𝑥 2 −√1−𝑥 2
2
[log(𝑥 + √𝑥 2 + 1)] ,
show that (1 + 𝑥 2 )
1−cos 4𝑥
,
{ 8𝑥 2
Find the value of 𝑘 for which 𝑓(𝑥) =
𝑑2 𝑦
𝑑𝑥 2
+𝑥
𝑥≠0
𝑘, 𝑥 = 0
𝑎 cos 𝑥
𝜋−2𝑥
15)
1
= 2𝑦−1
𝑑𝑦
√1+𝑥−√1−𝑥
), find 𝑑𝑥
√1+𝑥+√1−𝑥
2𝑥−3√1−𝑥 2
𝑑𝑦
cos−1 (
), find 𝑑𝑥
√13
1
𝑑𝑦
𝑥+
, then find 𝑑𝑥
1
𝑥+
1
12)
14)
√1+𝑥 2 −1
) w.r.
𝑥
𝑑𝑦
𝑑𝑥
Determine the values of 𝑎 and 𝑏, 𝑓(𝑥) = 3,
𝑏𝑡𝑎𝑛 2𝑥
{ (2𝑥−𝜋)
𝑑𝑦
𝑑𝑥
=2
is continuous at 𝑥 = 0
𝜋
, 𝑖𝑓 𝑥 < 2
𝑖𝑓 𝑥 =
, 𝑖𝑓 𝑥 >
𝜋
2
𝜋
2
is continuous at 𝑥 =
𝜋
2
16)
Show that the function 𝑓 defined as follows, is continuous at 𝑥 = 2, but not differentiable at
𝑥 = 2.
𝑓(𝑥) =
3𝑥 − 2, 𝑖𝑓 0 < 𝑥 ≤ 1
{ 2𝑥2 − 𝑥, 𝑖𝑓1 < 𝑥 ≤ 2
5𝑥 − 4 , 𝑖𝑓 𝑥 > 2
𝑎(1−sin 𝜋𝑥)
1
, 𝑖𝑓 𝑥 < 2
(1+cos 2𝜋𝑥)
17)
Let 𝑓(𝑥) =
𝑏,
𝑖𝑓 𝑥 =
√2𝑥−1
1
{√4+√2𝑥−1−2
1
2
, 𝑖𝑓 𝑥 >
Determine the sum of 𝑎 + 𝑏 so that 𝑓(𝑥) is continuous at 𝑥 =
1
2
.
2
18)
19)
Let𝑔(𝑥) =
1−𝑎𝑥 +𝑥𝑎𝑥 log 𝑎
,
𝑎 𝑥 𝑥2
0,
𝑖𝑓 𝑥 < 0
𝑖𝑓 𝑥 = 0
2𝑥 𝑎𝑥 −𝑥 log 2−𝑥 log 𝑎−1
𝑥2
{
is continuous at 𝑥 = 0, find the value of 𝑎
, 𝑖𝑓 𝑥 > 0