Differentiability for Competitive Exam Success

Differentiability

Q. For the function f(x) = x^2 - 2x + 1, find the slope of the tangent line at x = 1.

A. 0
B. 1
C. 2
D. 3

Solution

f'(x) = 2x - 2. Thus, f'(1) = 2(1) - 2 = 0.

Correct Answer: A — 0

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Q. For the function f(x) = x^2 - 4x + 4, find the point where it is not differentiable.

A. x = 0
B. x = 2
C. x = 4
D. It is differentiable everywhere

Solution

As a polynomial, f(x) is differentiable everywhere, including at x = 2.

Correct Answer: D — It is differentiable everywhere

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Q. For the function f(x) = x^3 - 3x^2 + 2, find the points where it is not differentiable.

A. None
B. x = 0
C. x = 1
D. x = 2

Solution

As a polynomial, f(x) is differentiable everywhere, hence no points of non-differentiability.

Correct Answer: A — None

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Q. For the function f(x) = x^3 - 3x^2 + 4, find the points where it is not differentiable.

A. None
B. x = 0
C. x = 1
D. x = 2

Solution

The function is a polynomial and is differentiable everywhere, hence there are no points where it is not differentiable.

Correct Answer: A — None

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Q. For the function f(x) = x^3 - 3x^2 + 4, find the value of x where f is not differentiable.

A. 0
B. 1
C. 2
D. 3

Solution

The function is a polynomial and is differentiable everywhere, so there is no such x.

Correct Answer: A — 0

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Q. For the function f(x) = x^3 - 3x^2 + 4, find the x-coordinate of the point where f is differentiable.

A. -1
B. 0
C. 1
D. 2

Solution

f(x) is a polynomial and is differentiable everywhere. The x-coordinate can be any real number.

Correct Answer: C — 1

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Q. For the function f(x) = { x^2, x < 1; kx + 1, x >= 1 }, find k such that f is differentiable at x = 1.

A. 0
B. 1
C. 2
D. 3

Solution

Setting f(1-) = f(1+) and f'(1-) = f'(1+) gives k = 2 for differentiability.

Correct Answer: B — 1

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Q. For the function f(x) = |x - 2| + |x + 3|, find the point where it is not differentiable.

A. -3
B. 2
C. 0
D. 1

Solution

The function is not differentiable at x = -3 and x = 2, but the first point of interest is -3.

Correct Answer: A — -3

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Q. For which value of a is the function f(x) = x^2 + ax + 1 differentiable at x = -1?

A. -1
B. 0
C. 1
D. 2

Solution

To ensure differentiability at x = -1, we find f'(-1) exists. Setting a = 0 ensures the derivative is defined.

Correct Answer: B — 0

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Q. For which value of a is the function f(x) = x^2 + ax + 1 differentiable everywhere?

A. -1
B. 0
C. 1
D. 2

Solution

The function is a polynomial and is differentiable for all real numbers, hence any value of a works.

Correct Answer: B — 0

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Q. For which value of a is the function f(x) = x^2 - ax + 2 differentiable at x = 1?

A. 0
B. 1
C. 2
D. 3

Solution

Setting the derivative f'(1) = 0 gives a = 1 for differentiability.

Correct Answer: B — 1

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Q. For which value of a is the function f(x) = x^2 - ax + 4 differentiable at x = 2?

A. 0
B. 2
C. 4
D. 6

Solution

f(x) is a polynomial and is differentiable for all a, hence any value of a works.

Correct Answer: A — 0

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Q. For which value of a is the function f(x) = x^3 - 3ax + 2 differentiable at x = 1?

A. 0
B. 1
C. 2
D. 3

Solution

Setting f'(1) = 0 gives a = 1, ensuring differentiability at that point.

Correct Answer: B — 1

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Q. For which value of a is the function f(x) = x^3 - 3ax^2 + 3a^2x + 1 differentiable at x = 1?

A. 0
B. 1
C. 2
D. 3

Solution

Setting f'(1) = 0 gives a = 1 for differentiability at x = 1.

Correct Answer: B — 1

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Q. For which value of k is the function f(x) = kx^2 + 2x + 1 differentiable at x = -1?

A. 0
B. 1
C. -1
D. 2

Solution

f'(x) = 2kx + 2; f'(-1) = -2k + 2 must exist for any k.

Correct Answer: A — 0

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Q. If f(x) = e^x, then f'(0) is equal to?

A. 0
B. 1
C. e
D. e^0

Solution

f'(x) = e^x; f'(0) = e^0 = 1.

Correct Answer: B — 1

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Q. If f(x) = ln(x) for x > 0, is f differentiable at x = 1?

A. Yes
B. No
C. Only continuous
D. Only left differentiable

Solution

f'(x) = 1/x; f'(1) = 1, hence f is differentiable at x = 1.

Correct Answer: A — Yes

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Q. If f(x) = x^2 + 2x + 1 for x < 0 and f(x) = kx + 1 for x >= 0, find k such that f is differentiable at x = 0.

A. -1
B. 0
C. 1
D. 2

Solution

Setting the left-hand derivative equal to the right-hand derivative at x = 0 gives k = 2.

Correct Answer: A — -1

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Q. If f(x) = x^2 + 2x + 1, find f'(1).

A. 1
B. 2
C. 3
D. 4

Solution

f'(x) = 2x + 2, thus f'(1) = 2(1) + 2 = 4.

Correct Answer: C — 3

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Q. If f(x) = x^2 + 2x + 1, what is f'(1)?

A. 2
B. 3
C. 4
D. 5

Solution

Calculating the derivative f'(x) = 2x + 2, we find f'(1) = 4.

Correct Answer: B — 3

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Q. If f(x) = x^2 + 2x + 3, find f'(1).

A. 2
B. 3
C. 4
D. 5

Solution

f'(x) = 2x + 2. Therefore, f'(1) = 2(1) + 2 = 4.

Correct Answer: C — 4

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Q. If f(x) = x^2 - 4x + 4, find f'(2).

A. 0
B. 1
C. 2
D. 4

Solution

f'(x) = 2x - 4. Thus, f'(2) = 2(2) - 4 = 0.

Correct Answer: A — 0

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Q. If f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1, is f differentiable at x = 1?

A. Yes
B. No
C. Only continuous
D. Only left differentiable

Solution

f'(1) from left = 2 and from right = 2; hence f is differentiable at x = 1.

Correct Answer: B — No

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Q. If f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0, is f differentiable at x = 0?

A. Yes
B. No
C. Only left differentiable
D. Only right differentiable

Solution

Using the limit definition of the derivative, f'(0) exists, hence f is differentiable at x = 0.

Correct Answer: A — Yes

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Q. If f(x) = x^3 - 3x + 2, find the critical points where f'(x) = 0.

A. -1
B. 0
C. 1
D. 2

Solution

Set f'(x) = 3x^2 - 3 = 0 and solve for x.

Correct Answer: B — 0

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Q. If f(x) = x^3 - 3x + 2, find the points where f is not differentiable.

A. 0
B. 1
C. 2
D. 3

Solution

The function is a polynomial and is differentiable everywhere, hence no points of non-differentiability.

Correct Answer: A — 0

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Q. If f(x) = x^3 - 3x^2 + 4, find the point where f is not differentiable.

A. 0
B. 1
C. 2
D. 3

Solution

The function is a polynomial and is differentiable everywhere, but checking critical points shows f'(x) = 0 at x = 2.

Correct Answer: C — 2

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Q. If f(x) = x^3 - 3x^2 + 4, then f'(1) is equal to?

A. 1
B. 2
C. 3
D. 4

Solution

f'(x) = 3x^2 - 6x; f'(1) = 3(1)^2 - 6(1) = 3 - 6 = -3.

Correct Answer: B — 2

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Q. If f(x) = x^3 - 3x^2 + 4, then f'(2) is equal to?

A. 0
B. 1
C. 2
D. 3

Solution

f'(x) = 3x^2 - 6x; f'(2) = 3(2^2) - 6(2) = 12 - 12 = 0.

Correct Answer: B — 1

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Q. If f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1, find f'(1).

A. 0
B. 1
C. 2
D. 3

Solution

Calculating f'(x) = 4x^3 - 12x^2 + 12x - 4. Thus, f'(1) = 4 - 12 + 12 - 4 = 0.

Correct Answer: A — 0

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Showing 31 to 60 of 91 (4 Pages)

Differentiability MCQ & Objective Questions

Differentiability is a crucial concept in calculus that plays a significant role in various examinations. Understanding this topic not only helps in grasping advanced mathematical concepts but also enhances your problem-solving skills. Practicing MCQs and objective questions on differentiability is essential for scoring better in your exams. By solving these practice questions, you can identify important questions and strengthen your exam preparation.

What You Will Practise Here

Definition and significance of differentiability
Conditions for differentiability at a point
Relationship between continuity and differentiability
Derivatives and their applications
Higher-order derivatives
Graphical interpretation of differentiable functions
Common differentiability problems and solutions

Exam Relevance

The topic of differentiability is frequently tested in CBSE, State Boards, NEET, and JEE examinations. Students can expect questions that require them to determine whether a function is differentiable at a given point or to apply the concept of derivatives in real-world scenarios. Common question patterns include multiple-choice questions that assess both theoretical understanding and practical application of differentiability concepts.

Common Mistakes Students Make

Confusing differentiability with continuity
Overlooking the conditions required for a function to be differentiable
Misinterpreting the graphical representation of differentiable functions
Neglecting higher-order derivatives in complex problems
Failing to apply the correct formulas in objective questions

FAQs

Question: What is the difference between continuity and differentiability?
Answer: Continuity means a function does not have any breaks or jumps, while differentiability indicates that a function has a defined derivative at a point.

Question: How can I determine if a function is differentiable at a point?
Answer: A function is differentiable at a point if it is continuous at that point and the limit of the difference quotient exists.

Question: Why is practicing MCQs on differentiability important?
Answer: Practicing MCQs helps reinforce your understanding, improves problem-solving speed, and prepares you for the types of questions you will encounter in exams.

Now is the time to enhance your understanding of differentiability! Dive into our practice MCQs and test your knowledge to ensure you are well-prepared for your upcoming exams.

Differentiability

Differentiability MCQ & Objective Questions

What You Will Practise Here

Exam Relevance

Common Mistakes Students Make

FAQs

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