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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Differentiability

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