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Principle: Differentiability at a point
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Q1
#1
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Moderate
Let
f
(
x
)
=
{
2
x
+
1
,
−
3
<
x
<
−
2
x
−
1
,
−
2
≤
x
<
0
x
+
2
,
0
≤
x
<
1
f(x)=\begin{cases}2x+1, & -3<x<-2\\x-1, & -2\leq x<0\\x+2, & 0\leq x<1\end{cases}
f
(
x
)
=
⎩
⎨
⎧
2
x
+
1
,
x
−
1
,
x
+
2
,
−
3
<
x
<
−
2
−
2
≤
x
<
0
0
≤
x
<
1
. Which one of the following statements is correct in respect of the above function?
Add
Lever: Differentiability at a point
A
It is discontinuous at
x
=
−
2
x=-2
x
=
−
2
but continuous at every other point.
B
It is continuous only in the interval
(
−
3
,
−
2
)
(-3,-2)
(
−
3
,
−
2
)
.
C
It is discontinuous at
x
=
0
x=0
x
=
0
but continuous at every other point.
D
It is discontinuous at every point.
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[Q92 · Apr · 2017]
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Q2
#2
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Moderate
Suppose the function
f
(
x
)
=
x
n
f(x)=x^n
f
(
x
)
=
x
n
,
n
≠
0
n\neq0
n
=
0
is differentiable for all
x
x
x
. Then
n
n
n
can be any element of the interval
Add
Lever: Differentiability at a point
A
[
1
,
∞
)
[1,\infty)
[
1
,
∞
)
B
(
0
,
∞
)
(0,\infty)
(
0
,
∞
)
C
(
1
2
,
∞
)
\left(\dfrac{1}{2},\infty\right)
(
2
1
,
∞
)
D
None of the above
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[Q99 · Apr · 2017]
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Q3
#3
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Easy
A function is defined as follows:
f
(
x
)
=
{
−
x
x
2
,
x
≠
0
0
,
x
=
0
f(x) = \begin{cases} -\dfrac{x}{\sqrt{x^2}}, & x \neq 0 \\ 0, & x = 0 \end{cases}
f
(
x
)
=
⎩
⎨
⎧
−
x
2
x
,
0
,
x
=
0
x
=
0
. Which one of the following is correct in respect of the above function?
Add
Lever: Differentiability at a point
A
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
0
x = 0
x
=
0
but not differentiable at
x
=
0
x = 0
x
=
0
B
f
(
x
)
f(x)
f
(
x
)
is continuous as well as differentiable at
x
=
0
x = 0
x
=
0
C
f
(
x
)
f(x)
f
(
x
)
is discontinuous at
x
=
0
x = 0
x
=
0
D
None of the above
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[Q62 · Sep · 2017]
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Q4
#4
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Moderate
A function is defined in
(
0
,
∞
)
(0, \infty)
(
0
,
∞
)
by
f
(
x
)
=
{
1
−
x
2
for
0
<
x
≤
1
ln
x
for
1
<
x
≤
2
ln
2
−
1
+
0.5
x
for
2
<
x
<
∞
f(x) = \begin{cases}1 - x^2 & \text{for } 0 < x \leq 1 \\ \ln x & \text{for } 1 < x \leq 2 \\ \ln 2 - 1 + 0.5x & \text{for } 2 < x < \infty\end{cases}
f
(
x
)
=
⎩
⎨
⎧
1
−
x
2
ln
x
ln
2
−
1
+
0.5
x
for
0
<
x
≤
1
for
1
<
x
≤
2
for
2
<
x
<
∞
. Which one of the following is correct in respect of the derivative of the function, i.e.,
f
′
(
x
)
f'(x)
f
′
(
x
)
?
Add
Lever: Differentiability at a point
A
f
′
(
x
)
=
2
x
f'(x) = 2x
f
′
(
x
)
=
2
x
for
0
<
x
≤
1
0 < x \leq 1
0
<
x
≤
1
B
f
′
(
x
)
=
−
2
x
f'(x) = -2x
f
′
(
x
)
=
−
2
x
for
0
<
x
≤
1
0 < x \leq 1
0
<
x
≤
1
C
f
′
(
x
)
=
−
2
x
f'(x) = -2x
f
′
(
x
)
=
−
2
x
for
0
<
x
<
1
0 < x < 1
0
<
x
<
1
D
f
′
(
x
)
=
0
f'(x) = 0
f
′
(
x
)
=
0
for
0
<
x
<
∞
0 < x < \infty
0
<
x
<
∞
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[Q68 · Sep · 2017]
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Q5
#5
NDA → Mathematics → Limits & Continuity → One-Sided Limits, Greatest Integer, and Absolute Value Limits
·
Hard
The left-hand derivative of
f
(
x
)
=
[
x
]
sin
(
π
x
)
f(x) = [x]\sin(\pi x)
f
(
x
)
=
[
x
]
sin
(
π
x
)
at
x
=
k
x = k
x
=
k
, where
k
k
k
is an integer and
[
x
]
[x]
[
x
]
is the greatest integer function, is
Add
Lever: Differentiability at a point
A
(
−
1
)
k
(
k
−
1
)
π
(-1)^k(k-1)\pi
(
−
1
)
k
(
k
−
1
)
π
B
(
−
1
)
k
−
1
(
k
−
1
)
π
(-1)^{k-1}(k-1)\pi
(
−
1
)
k
−
1
(
k
−
1
)
π
C
(
−
1
)
k
k
π
(-1)^k k\pi
(
−
1
)
k
k
π
D
(
−
1
)
k
−
1
k
π
(-1)^{k-1}k\pi
(
−
1
)
k
−
1
k
π
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[Q79 · Sep · 2017]
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Q6
#6
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Hard
The set of all points, where the function
f
(
x
)
=
1
−
e
−
x
2
f(x) = \sqrt{1 - e^{-x^2}}
f
(
x
)
=
1
−
e
−
x
2
is differentiable, is
Add
Lever: Differentiability at a point
A
(
0
,
∞
)
(0, \infty)
(
0
,
∞
)
B
(
−
∞
,
∞
)
(-\infty, \infty)
(
−
∞
,
∞
)
C
(
−
∞
,
0
)
∪
(
0
,
∞
)
(-\infty, 0) \cup (0, \infty)
(
−
∞
,
0
)
∪
(
0
,
∞
)
D
(
−
1
,
∞
)
(-1, \infty)
(
−
1
,
∞
)
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[Q83 · Sep · 2017]
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Q7
#7
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Easy
If
f
(
x
)
=
∣
x
∣
+
∣
x
−
1
∣
f(x)=|x|+|x-1|
f
(
x
)
=
∣
x
∣
+
∣
x
−
1∣
, which is correct?
Add
Lever: Differentiability at a point
A
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
0
x=0
x
=
0
and
x
=
1
x=1
x
=
1
B
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
0
x=0
x
=
0
but not at
x
=
1
x=1
x
=
1
C
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
1
x=1
x
=
1
but not at
x
=
0
x=0
x
=
0
D
f
(
x
)
f(x)
f
(
x
)
is neither continuous at
x
=
0
x=0
x
=
0
nor at
x
=
1
x=1
x
=
1
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[Q73 · Apr · 2018]
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Q8
#8
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Moderate
For
f
(
x
)
=
x
2
ln
∣
x
∣
f(x)=x^2\ln|x|
f
(
x
)
=
x
2
ln
∣
x
∣
(
x
≠
0
x\ne0
x
=
0
),
f
(
0
)
=
0
f(0)=0
f
(
0
)
=
0
, what is
f
′
(
0
)
f'(0)
f
′
(
0
)
?
Add
Lever: Differentiability at a point
A
0
B
1
C
-1
D
It does not exist
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[Q74 · Apr · 2018]
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Q9
#9
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Easy
For the function
f
(
x
)
=
∣
x
−
3
∣
f(x)=|x-3|
f
(
x
)
=
∣
x
−
3∣
, which one of the following is
not
\textbf{\text{not}}
not
correct?
Add
Lever: Differentiability at a point
A
The function is not continuous at x = -3
B
The function is continuous at x = 3
C
The function is differentiable at x = 0
D
The function is differentiable at x = -3
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[Q74 · Sep · 2018]
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Q10
#10
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Moderate
Consider the following statements for
f
(
x
)
=
e
−
∣
x
∣
f(x)=e^{-|x|}
f
(
x
)
=
e
−
∣
x
∣
: 1. The function is continuous at
x
=
0
x=0
x
=
0
. 2. The function is differentiable at
x
=
0
x=0
x
=
0
. Which of the above statements is/are correct?
Add
Lever: Differentiability at a point
A
1 only
B
2 only
C
Both 1 and 2
D
Neither 1 nor 2
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[Q91 · Apr · 2020]
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Q11
#11
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Easy
If
f
(
x
)
=
e
∣
x
∣
f(x)=e^{|x|}
f
(
x
)
=
e
∣
x
∣
, then which one of the following is correct?
Add
Lever: Differentiability at a point
A
f
′
(
0
)
=
1
f'(0)=1
f
′
(
0
)
=
1
B
f
′
(
0
)
=
−
1
f'(0)=-1
f
′
(
0
)
=
−
1
C
f
′
(
0
)
=
0
f'(0)=0
f
′
(
0
)
=
0
D
f
′
(
0
)
f'(0)
f
′
(
0
)
does not exist
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Show solution
[Q84 · Apr · 2021]
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Q12
#12
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Easy
Consider the following statements in respect of
f
(
x
)
=
∣
x
∣
−
1
f(x)=|x|-1
f
(
x
)
=
∣
x
∣
−
1
: 1.
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
1
x=1
x
=
1
. 2.
f
(
x
)
f(x)
f
(
x
)
is differentiable at
x
=
0
x=0
x
=
0
. Which of the above statements is/are correct?
Add
Lever: Differentiability at a point
A
1 only
B
2 only
C
Both 1 and 2
D
Neither 1 nor 2
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[Q76 · Sep · 2021]
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Q13
#13
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Easy
Let
y
=
[
x
+
1
]
y=[x+1]
y
=
[
x
+
1
]
,
−
4
<
x
<
−
3
-4<x<-3
−
4
<
x
<
−
3
where
[
.
]
[.]
[
.
]
is the greatest integer function. What is the derivative of
y
y
y
with respect to
x
x
x
at
x
=
−
3.5
x=-3.5
x
=
−
3.5
?
Add
Lever: Differentiability at a point
A
-4
B
-3.5
C
-3
D
0
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[Q71 · Apr · 2022]
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Q14
#14
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Easy
If
f
(
x
)
f(x)
f
(
x
)
is differentiable at
x
=
a
x=a
x
=
a
, then consider the following statements: I.
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
a
x=a
x
=
a
. II.
lim
x
→
a
f
(
x
)
=
f
(
a
)
\lim_{x\to a}f(x)=f(a)
lim
x
→
a
f
(
x
)
=
f
(
a
)
. Which of the statements given above is/are correct?
Add
Lever: Differentiability at a point
A
I only
B
II only
C
Both I and II
D
Neither I nor II
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[Q71 · Apr · 2026]
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Set · 2 questions
For the next two (02) items that follow: Let
f
(
x
)
=
{
a
x
(
x
−
1
)
,
x
<
1
x
−
1
,
1
≤
x
≤
3
p
x
2
+
q
x
+
2
,
x
>
3
f(x)=\begin{cases}ax(x-1), & x<1\\ x-1, & 1\leq x\leq3\\ px^2+qx+2, & x>3\end{cases}
f
(
x
)
=
⎩
⎨
⎧
a
x
(
x
−
1
)
,
x
−
1
,
p
x
2
+
q
x
+
2
,
x
<
1
1
≤
x
≤
3
x
>
3
. Given that
f
(
x
)
f(x)
f
(
x
)
is continuous for all x but not differentiable at x=1. Further
f
′
(
x
)
f'(x)
f
′
(
x
)
is continuous at x=3.
Q15
#15
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Hard
What is the value of p?
Add
Lever: Differentiability at a point
A
-1
B
−
1
3
-\dfrac{1}{3}
−
3
1
C
1
3
\dfrac{1}{3}
3
1
D
1
Tap an option to check your answer.
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[Q95 · Apr · 2026]
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Q16
#16
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Easy
What is the value of q?
Add
Lever: Differentiability at a point
A
-1
B
−
1
3
-\dfrac{1}{3}
−
3
1
C
1
3
\dfrac{1}{3}
3
1
D
1
Tap an option to check your answer.
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[Q96 · Apr · 2026]
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Set · 2 questions
For the next two (02) items that follow: Let
f
(
x
)
=
tan
(
x
2
)
f(x)=\tan(x^2)
f
(
x
)
=
tan
(
x
2
)
and
g
(
x
)
=
x
∣
x
∣
g(x)=x|x|
g
(
x
)
=
x
∣
x
∣
for
∣
x
∣
<
π
/
2
|x|<\sqrt{\pi/2}
∣
x
∣
<
π
/2
.
Q17
#17
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Moderate
If
p
(
x
)
=
f
(
x
)
g
(
x
)
p(x)=f(x)g(x)
p
(
x
)
=
f
(
x
)
g
(
x
)
, then which of the following statements is/are correct? I.
p
(
x
)
p(x)
p
(
x
)
is continuous at
x
=
0
x=0
x
=
0
. II.
p
(
x
)
p(x)
p
(
x
)
is differentiable at
x
=
0
x=0
x
=
0
. Select the answer using the code given below.
Add
Lever: Differentiability at a point
A
I only
B
II only
C
Both I and II
D
Neither I nor II
Tap an option to check your answer.
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[Q99 · Apr · 2026]
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Q18
#18
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Moderate
If
q
(
x
)
=
f
∘
g
(
x
)
q(x)=f\circ g(x)
q
(
x
)
=
f
∘
g
(
x
)
, then which of the following statements is/are correct? I.
q
(
x
)
q(x)
q
(
x
)
is continuous at
x
=
0
x=0
x
=
0
. II.
q
(
x
)
q(x)
q
(
x
)
is differentiable at
x
=
0
x=0
x
=
0
. Select the answer using the code given below.
Add
Lever: Differentiability at a point
A
I only
B
II only
C
Both I and II
D
Neither I nor II
Tap an option to check your answer.
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[Q100 · Apr · 2026]
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Set · 1 question
For the following three (03) items: Consider the function
f
(
x
)
=
x
∣
x
∣
f(x)=x|x|
f
(
x
)
=
x
∣
x
∣
.
Q19
#19
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Moderate
Consider the following statements: I. The function is increasing in the interval
(
−
∞
,
∞
)
(-\infty,\infty)
(
−
∞
,
∞
)
. II. The function is differentiable at
x
=
0
x=0
x
=
0
. Which of the statements given above is/are correct?
Add
Lever: Differentiability at a point
A
I only
B
II only
C
Both I and II
D
Neither I nor II
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[Q75 · Sep · 2025]
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Set · 2 questions
For the following two (02) items: Let the function
f
(
x
)
=
∣
x
−
3
∣
+
∣
x
−
4
∣
f(x)=|x-3|+|x-4|
f
(
x
)
=
∣
x
−
3∣
+
∣
x
−
4∣
be defined on the interval
[
0
,
5
]
[0,5]
[
0
,
5
]
.
Q20
#20
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Easy
What is
d
y
d
x
\dfrac{dy}{dx}
d
x
d
y
at
x
=
3.5
x=3.5
x
=
3.5
equal to?
Add
Lever: Differentiability at a point
A
0
B
1
C
2
D
3.5
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[Q87 · Sep · 2025]
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Q21
#21
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Moderate
Consider the following statements: I. The function is differentiable at
x
=
3
x=3
x
=
3
. II. The function is differentiable at
x
=
4
x=4
x
=
4
. Which of the statements given above is/are correct?
Add
Lever: Differentiability at a point
A
I only
B
II only
C
Both I and II
D
Neither I nor II
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[Q88 · Sep · 2025]
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Set · 1 question
Let
f
(
x
)
=
a
x
x
+
1
+
b
,
x
<
1
f(x)=\frac{ax}{x+1}+b,\ x<1
f
(
x
)
=
x
+
1
a
x
+
b
,
x
<
1
and
x
−
1
,
1
≤
x
≤
2
\sqrt{x-1},\ 1\leq x\leq2
x
−
1
,
1
≤
x
≤
2
.
Q22
#22
NDA → Mathematics → Differentiation → Differentiability of Absolute Value, Piecewise, and Greatest Integer Functions
·
Hard
If
f
(
x
)
f(x)
f
(
x
)
is differentiable at
x
=
1
x=1
x
=
1
, then what is the value of
(
a
+
b
)
(a+b)
(
a
+
b
)
?
Add
Lever: Differentiability at a point
A
−
1
3
-\frac{1}{3}
−
3
1
B
-1
C
0
D
1
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[Q63 · Sep · 2023]
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Q23
#23
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Moderate
Let
f
(
x
)
=
∣
x
∣
+
1
f(x)=|x|+1
f
(
x
)
=
∣
x
∣
+
1
and
g
(
x
)
=
[
x
]
−
1
g(x)=[x]-1
g
(
x
)
=
[
x
]
−
1
, where
[
⋅
]
[\cdot]
[
⋅
]
is the greatest integer function. Let
h
(
x
)
=
f
(
x
)
⋅
g
(
x
)
h(x)=f(x)\cdot g(x)
h
(
x
)
=
f
(
x
)
⋅
g
(
x
)
. Consider the following statements: (A)
f
(
x
)
f(x)
f
(
x
)
is differentiable for all
x
<
0
x<0
x
<
0
. (B)
g
(
x
)
g(x)
g
(
x
)
is continuous at
x
=
0.0001
x=0.0001
x
=
0.0001
. (C) The derivative of
g
(
x
)
g(x)
g
(
x
)
at
x
=
2.5
x=2.5
x
=
2.5
is 1. Which of the statements given above are correct?
Add
Lever: Differentiability at a point
A
A and B only
B
B and C only
C
A and C only
D
A, B and C
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[Q87 · Apr · 2024]
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Set · 1 question
Let
f
(
x
)
=
{
x
3
,
x
2
<
1
x
2
,
x
2
≥
1
f(x) = \begin{cases} x^3, & x^2 < 1 \\ x^2, & x^2 \geq 1 \end{cases}
f
(
x
)
=
{
x
3
,
x
2
,
x
2
<
1
x
2
≥
1
Q24
#24
NDA → Mathematics → Limits & Continuity → Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory
·
Moderate
Consider the following statements: (I). The function is continuous at
x
=
−
1
x = -1
x
=
−
1
. (II). The function is differentiable at
x
=
1
x = 1
x
=
1
. Which of the statements given above is/are correct?
Add
Lever: Differentiability at a point
A
I only
B
II only
C
Both I and II
D
Neither I nor II
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[Q80 · Apr · 2025]
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