Differentiability for Competitive Exam Success

Differentiability

Q. Determine if the function f(x) = x^3 - 3x + 2 is differentiable at x = 1.

A. Yes
B. No
C. Only from the left
D. Only from the right

Solution

f(x) is a polynomial function, which is differentiable everywhere, including at x = 1.

Correct Answer: A — Yes

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Q. Determine if the function f(x) = |x - 1| is differentiable at x = 1.

A. Yes
B. No
C. Only from the left
D. Only from the right

Solution

The left-hand derivative is -1 and the right-hand derivative is 1. Since they are not equal, f(x) is not differentiable at x = 1.

Correct Answer: B — No

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Q. Determine the point at which the function f(x) = |x - 1| is not differentiable.

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

Solution

The function |x - 1| is not differentiable at x = 1 due to a cusp.

Correct Answer: B — x = 1

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Q. Determine the point at which the function f(x) = |x - 3| is not differentiable.

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

Solution

The function f(x) = |x - 3| is not differentiable at x = 3 because it has a sharp corner.

Correct Answer: C — x = 3

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Q. Determine the point at which the function f(x) = |x^2 - 4| is differentiable.

A. x = -2
B. x = 0
C. x = 2
D. x = -4

Solution

f(x) is not differentiable at x = -2 and x = 2, but is differentiable everywhere else.

Correct Answer: A — x = -2

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Q. Determine the points where f(x) = x^3 - 3x is not differentiable.

A. x = 0
B. x = 1
C. x = -1
D. Nowhere

Solution

The function is a polynomial and is differentiable everywhere, hence nowhere.

Correct Answer: D — Nowhere

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Q. Determine the points where the function f(x) = x^4 - 4x^3 is not differentiable.

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

Solution

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

Correct Answer: D — None

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Q. Determine the value of a for which the function f(x) = { x^2 + a, x < 1; 2x + 3, x >= 1 } is differentiable at x = 1.

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

Solution

Setting f(1-) = f(1+) and f'(1-) = f'(1+) leads to a = 1 for differentiability.

Correct Answer: B — 0

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Q. Determine the value of m for which the function f(x) = { mx + 1, x < 2; x^2 - 4, x >= 2 } is differentiable at x = 2.

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

Solution

Setting f(2-) = f(2+) and f'(2-) = f'(2+) leads to m = 1 for differentiability.

Correct Answer: B — 0

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Q. Evaluate the derivative of f(x) = e^x + ln(x) at x = 1.

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

Solution

f'(x) = e^x + 1/x. At x = 1, f'(1) = e + 1.

Correct Answer: A — 1

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Q. Find the derivative of f(x) = e^(2x) at x = 0.

A. 1
B. 2
C. e
D. 2e

Solution

f'(x) = 2e^(2x). At x = 0, f'(0) = 2e^0 = 2.

Correct Answer: B — 2

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Q. Find the derivative of f(x) = e^(x^2).

A. 2xe^(x^2)
B. e^(x^2)
C. x e^(x^2)
D. 2e^(x^2)

Solution

Using the chain rule, f'(x) = e^(x^2) * 2x = 2x e^(x^2).

Correct Answer: A — 2xe^(x^2)

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Q. Find the derivative of f(x) = e^x * ln(x) at x = 1.

A. 1
B. 0
C. e
D. ln(e)

Solution

Using the product rule, f'(x) = e^x * ln(x) + e^x/x. At x = 1, this simplifies to 0.

Correct Answer: A — 1

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Q. Find the derivative of f(x) = e^x * sin(x) at x = 0.

A. 1
B. 0
C. e
D. sin(0)

Solution

Using the product rule, f'(0) = e^0 * sin(0) + e^0 * cos(0) = 0 + 1 = 1.

Correct Answer: A — 1

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Q. Find the derivative of f(x) = ln(x^2 + 1) at x = 1.

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

Solution

f'(x) = (2x)/(x^2 + 1). At x = 1, f'(1) = (2*1)/(1^2 + 1) = 2/2 = 1.

Correct Answer: B — 1

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Q. Find the derivative of f(x) = sin(x) + cos(x) at x = π/4.

A. 0
B. 1
C. √2
D. √2/2

Solution

f'(x) = cos(x) - sin(x), thus f'(π/4) = √2/2 - √2/2 = 0.

Correct Answer: C — √2

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Q. Find the derivative of f(x) = tan(x) at x = π/4.

A. 1
B. 2
C. √2
D. 0

Solution

f'(x) = sec^2(x). At x = π/4, f'(π/4) = sec^2(π/4) = 2.

Correct Answer: A — 1

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Q. Find the derivative of f(x) = x^2 sin(1/x) at x = 0.

A. 0
B. 1
C. undefined
D. does not exist

Solution

Using the limit definition of the derivative, we find that f'(0) = 0, hence it is differentiable at x = 0.

Correct Answer: A — 0

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Q. Find the derivative of f(x) = x^3 - 3x^2 + 4 at x = 2.

A. 6
B. 8
C. 10
D. 12

Solution

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

Correct Answer: B — 8

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Q. Find the second derivative of f(x) = e^x at x = 0.

A. 0
B. 1
C. e
D. e^2

Solution

f''(x) = e^x, thus f''(0) = e^0 = 1.

Correct Answer: B — 1

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Q. Find the value of a for which the function f(x) = { ax + 1, x < 2; x^2 - 4, x >= 2 } is differentiable at x = 2.

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

Solution

Set the left-hand limit equal to the right-hand limit and their derivatives at x = 2.

Correct Answer: B — 1

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Q. Find the value of c such that the function f(x) = { x^2 + c, x < 1; 2x + 1, x >= 1 } is differentiable at x = 1.

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

Solution

Setting the left-hand limit equal to the right-hand limit gives c = 1.

Correct Answer: B — 1

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Q. Find the value of k for which the function f(x) = kx^2 + 2x + 1 is differentiable at x = 0.

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

Solution

f'(x) = 2kx + 2. At x = 0, f'(0) = 2. The function is differentiable for any k, but k = 0 gives a constant function.

Correct Answer: A — 0

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Q. Find the value of k for which the function f(x) = kx^2 + 3x + 2 is differentiable everywhere.

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

Solution

The function is a polynomial and is differentiable for all k, hence k can be any real number.

Correct Answer: A — k = 0

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Q. Find the value of k for which the function f(x) = x^3 - 3kx^2 + 3k^2x - k^3 is differentiable at x = k.

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

Solution

For f(x) to be differentiable at x = k, f'(k) must exist. Setting k = 1 makes f'(k) continuous.

Correct Answer: B — 1

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Q. For f(x) = { x^2, x < 1; 2x - 1, x ≥ 1 }, is f differentiable at x = 1?

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

Solution

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

Correct Answer: B — No

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Q. For the function f(x) = ln(x), find the point where it is not differentiable.

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

Solution

f(x) = ln(x) is not defined for x ≤ 0, hence not differentiable at x = 0.

Correct Answer: A — x = 0

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

A. 2x + 1
B. 2x + 2
C. 2x
D. x + 1

Solution

f'(x) = 2x + 2.

Correct Answer: B — 2x + 2

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

A. x = -1
B. x = 0
C. x = 1
D. It is differentiable everywhere

Solution

The function is a polynomial and is differentiable everywhere.

Correct Answer: D — It is differentiable everywhere

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Q. For the function f(x) = x^2 + kx + 1 to be differentiable at x = -1, what must k be?

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

Solution

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

Correct Answer: C — 1

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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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