Q. If f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1, find f'(2).
Solution
f'(x) = 4x^3 - 12x^2 + 12x - 4; f'(2) = 0.
Correct Answer: A — 0
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Q. If f(x) = x^4 - 4x^3 + 6x^2, find f'(2).
Solution
f'(x) = 4x^3 - 12x^2 + 12x; f'(2) = 4(2^3) - 12(2^2) + 12(2) = 32 - 48 + 24 = 8.
Correct Answer: A — 0
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Q. If f(x) = { x^2, x < 0; kx + 1, x >= 0 } is differentiable at x = 0, what is k?
Solution
Setting the derivatives equal at x = 0 gives k = 0.
Correct Answer: B — 0
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Q. If f(x) = |x|, is f differentiable at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only from the right
-
D.
Only from the left
Solution
The left and right derivatives at x = 0 do not match, hence f is not differentiable at that point.
Correct Answer: B — No
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Q. Is the function f(x) = x^2 - 2x + 1 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. Is the function f(x) = x^2 - 4x + 4 differentiable at x = 2?
-
A.
Yes
-
B.
No
-
C.
Only from the left
-
D.
Only from the right
Solution
The function is a polynomial and is differentiable everywhere, hence yes.
Correct Answer: A — Yes
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Q. Is the function f(x) = x^2 - 4x + 4 differentiable everywhere?
-
A.
Yes
-
B.
No
-
C.
Only at x = 0
-
D.
Only at x = 2
Solution
This is a polynomial function, which is differentiable everywhere on its domain.
Correct Answer: A — Yes
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Q. Is the function f(x) = x^2 sin(1/x) differentiable at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only from the left
-
D.
Only from the right
Solution
Using the limit definition, f'(0) = lim (h -> 0) [(h^2 sin(1/h) - 0)/h] = 0. Thus, f(x) is differentiable at x = 0.
Correct Answer: A — Yes
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Q. Is the function f(x) = x^3 - 3x + 2 differentiable at x = 1?
-
A.
Yes
-
B.
No
-
C.
Only left differentiable
-
D.
Only right differentiable
Solution
The function is a polynomial and hence differentiable everywhere, including at x = 1.
Correct Answer: A — Yes
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Q. The function f(x) = e^x is differentiable at all points?
-
A.
True
-
B.
False
-
C.
Only at x = 0
-
D.
Only at x = 1
Solution
f(x) = e^x is differentiable everywhere as it is an exponential function.
Correct Answer: A — True
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Q. The function f(x) = ln(x) is differentiable at x = 1?
-
A.
Yes
-
B.
No
-
C.
Only for x > 1
-
D.
Only for x < 1
Solution
f'(x) = 1/x; f'(1) = 1/1 = 1, hence it is differentiable at x = 1.
Correct Answer: A — Yes
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Q. The function f(x) = sqrt(x) is differentiable at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only from the right
-
D.
Only from the left
Solution
f(x) = sqrt(x) is not differentiable at x = 0 because the left-hand derivative does not exist.
Correct Answer: B — No
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Q. The function f(x) = x^2 + 2x + 1 is differentiable everywhere?
-
A.
True
-
B.
False
-
C.
Only at x = 0
-
D.
Only for x > 0
Solution
f(x) is a polynomial function, which is differentiable everywhere.
Correct Answer: A — True
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Q. The function f(x) = x^2 - 2x + 1 is differentiable at all points?
-
A.
True
-
B.
False
-
C.
Only at x = 0
-
D.
Only for x > 0
Solution
f(x) is a polynomial function, which is differentiable everywhere.
Correct Answer: A — True
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Q. The function f(x) = x^2 - 2x + 1 is differentiable at x = 2?
-
A.
Yes
-
B.
No
-
C.
Only left
-
D.
Only right
Solution
f(x) is a polynomial function, hence it is differentiable everywhere including at x = 2.
Correct Answer: A — Yes
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Q. The function f(x) = x^2 - 4x + 4 is differentiable at x = 2?
-
A.
Yes
-
B.
No
-
C.
Only left
-
D.
Only right
Solution
f(x) is a polynomial function, hence differentiable everywhere including at x = 2.
Correct Answer: A — Yes
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Q. The function f(x) = x^2 - 4x + 4 is differentiable everywhere?
-
A.
True
-
B.
False
-
C.
Only at x = 0
-
D.
Only at x = 2
Solution
f(x) is a polynomial function, hence it is differentiable everywhere.
Correct Answer: A — True
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Q. The function f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1 is differentiable at x = 1?
-
A.
Yes
-
B.
No
-
C.
Only continuous
-
D.
Only from the left
Solution
f'(1) from left = 2 and from right = 2; hence, f is continuous but not differentiable at x = 1.
Correct Answer: B — No
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Q. The function f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0 is differentiable at x = 0. True or False?
-
A.
True
-
B.
False
-
C.
Depends on x
-
D.
Not enough information
Solution
True, as the limit of f'(x) as x approaches 0 exists and equals 0.
Correct Answer: A — True
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Q. The function f(x) = x^3 - 3x + 2 is differentiable at x = 1?
-
A.
Yes
-
B.
No
-
C.
Only left
-
D.
Only right
Solution
f(x) is a polynomial function, hence it is differentiable everywhere including at x = 1.
Correct Answer: A — Yes
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Q. The function f(x) = x^3 - 3x + 2 is differentiable everywhere. Find its critical points.
Solution
f'(x) = 3x^2 - 3 = 0 gives x = ±1, thus critical points are x = -1 and x = 1.
Correct Answer: B — 0
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Q. The function f(x) = x^3 - 3x + 2 is differentiable everywhere. What is f'(1)?
Solution
f'(x) = 3x^2 - 3, thus f'(1) = 0.
Correct Answer: A — 0
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Q. The function f(x) = { x^2, x < 1; 2x - 1, x ≥ 1 } is differentiable at x = 1 if which condition holds?
-
A.
f(1) = 1
-
B.
f'(1) = 1
-
C.
f'(1) = 2
-
D.
f(1) = 2
Solution
For differentiability, the left and right derivatives must equal at x = 1, hence f'(1) = 1.
Correct Answer: B — f'(1) = 1
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Q. The function f(x) = |x| is differentiable at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only from the right
-
D.
Only from the left
Solution
f(x) = |x| is not differentiable at x = 0 because the left-hand and right-hand derivatives do not match.
Correct Answer: B — No
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Q. What is the derivative of f(x) = ln(x^2 + 1) at x = 0?
-
A.
0
-
B.
1
-
C.
2
-
D.
undefined
Solution
f'(x) = (2x)/(x^2 + 1), thus f'(0) = 0.
Correct Answer: A — 0
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Q. What is the derivative of f(x) = ln(x^2 + 1) at x = 1?
Solution
f'(x) = (2x)/(x^2 + 1). At x = 1, f'(1) = (2*1)/(1^2 + 1) = 1.
Correct Answer: B — 1
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Q. What is the derivative of f(x) = sin(x^2)?
-
A.
2x cos(x^2)
-
B.
cos(x^2)
-
C.
2x sin(x^2)
-
D.
sin(x^2)
Solution
Using the chain rule, f'(x) = cos(x^2) * 2x = 2x cos(x^2).
Correct Answer: A — 2x cos(x^2)
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Q. What is the derivative of f(x) = |x| at x = 0?
-
A.
0
-
B.
1
-
C.
-1
-
D.
Undefined
Solution
The left-hand derivative is -1 and the right-hand derivative is 1, hence the derivative at x = 0 is undefined.
Correct Answer: D — Undefined
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Q. Which of the following functions is differentiable at x = 1? f(x) = { x^2, x < 1; 2x - 1, x >= 1 }
-
A.
f(1) = 1
-
B.
f(1) = 0
-
C.
f(1) = 2
-
D.
f(1) = 3
Solution
Check continuity and differentiability at x = 1 by equating left and right derivatives.
Correct Answer: A — f(1) = 1
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Q. Which of the following functions is differentiable everywhere?
-
A.
f(x) =
-
B.
x
-
C.
-
D.
f(x) = x^2
-
.
f(x) = sqrt(x)
-
.
f(x) = 1/x
Solution
f(x) = x^2 is a polynomial and differentiable everywhere.
Correct Answer: B — x
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