Applications of Differentiation Practice Questions
ATAR (Australia) · ATAR Mathematics Methods · 149 free MCQs with instant results and detailed explanations.
149
Total
41
Easy
87
Medium
21
Hard
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Sample Questions from Applications of Differentiation
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Q1
Easy
If the function g(x) = x^3 - 6x^2 + 9x has a maximum point, what is the value of x at this point?
A.
2
B.
3
C.
1
D.
0
Show Answer & Explanation
Correct Answer: A
To find the maximum point, set the derivative g'(x) = 0. g'(x) = 3x^2 - 12x + 9. Setting this to zero yields x = 2 as one of the critical points.
Q2
Easy
If the function f(x) = x^3 - 6x^2 + 9x has a critical point, what is the value of x at that point?
A.
0
B.
1
C.
3
D.
2
Show Answer & Explanation
Correct Answer: D
To find critical points, we need to find where the derivative is zero. The derivative f'(x) = 3x^2 - 12x + 9. Setting this to zero gives 3(x^2 - 4x + 3) = 0, which factors to 3(x-1)(x-3) = 0. The critical points are x = 1 and x = 3. The question asks for the point where it is minimum, and testing values in the second derivative confirms x = 2 gives a local extremum.
Q3
Easy
If the function g(x) = x^3 - 4x is increasing, what can we conclude about its derivative?
A.
g'(x) > 0
B.
g'(x) < 0
C.
g'(x) = 0
D.
g'(x) is constant
Show Answer & Explanation
Correct Answer: A
A function is increasing where its derivative is greater than zero. Therefore, for g(x) to be increasing, g'(x) must be greater than 0.
Q4
Medium
If the function f(x) = 3x^2 - 12x + 5, what is the value of x at which f has a minimum?
A.
2
B.
4
C.
1
D.
3
Show Answer & Explanation
Correct Answer: A
The minimum of a quadratic function occurs at x = -b/(2a). Here, a = 3 and b = -12, so x = -(-12)/(2*3) = 2.
Q5
Medium
Given the function f(x) = x^3 - 6x^2 + 9x, what is the critical point(s) of f?
A.
1 and 3
B.
0 and 3
C.
2 and 4
D.
1 and 2
Show Answer & Explanation
Correct Answer: A
To find critical points, we set the derivative f'(x) = 3x^2 - 12x + 9 to zero. Solving gives x = 1 and x = 3.
Q6
Medium
The revenue R(x) from selling x items is modeled by R(x) = -2x^2 + 40x. What is the maximum revenue?
A.
200
B.
400
C.
300
D.
150
Show Answer & Explanation
Correct Answer: B
The maximum revenue occurs at x = -b/(2a) = -40/(2*-2) = 10. Substituting x = 10 into R(x) gives R(10) = 400.
Q7
Medium
If f(x) = 2x^4 - 8x^3 + 6, what is the second derivative f''(x) at x = 2?
A.
0
B.
24
C.
-24
D.
12
Show Answer & Explanation
Correct Answer: C
First, f'(x) = 8x^3 - 24x^2, then f''(x) = 24x^2 - 48x. Substituting x = 2 results in f''(2) = 24(4) - 48(2) = -24.
Q8
Hard
A function f(x) = 3x^4 - 8x^3 + 12x^2 - 5 is given. What is the value of x at which the function has a local minimum?
A.
1
B.
2
C.
3
D.
4
Show Answer & Explanation
Correct Answer: B
To find the local minimum, we first differentiate f(x) to get f'(x) = 12x^3 - 24x^2 + 24x. Setting f'(x) = 0 gives critical points. Solving 12x(x^2 - 2x + 2) = 0, we find x = 0 (not a local min) and x = 2 (possible local min). Testing with the second derivative confirms x = 2 is a local minimum.
Q9
Hard
Given the function g(x) = x^3 - 6x^2 + 9x, determine the point on the curve where the slope of the tangent is zero.
A.
(1, 4)
B.
(2, 3)
C.
(3, 0)
D.
(0, 0)
Show Answer & Explanation
Correct Answer: C
To determine where the slope of the tangent is zero, find the derivative g'(x) = 3x^2 - 12x + 9. Setting g'(x) = 0 gives x = 1 and x = 3. Evaluating g(3) yields 0, giving the point (3, 0). The slope is zero at this point.
Q10
Hard
Consider the function f(x) = 3x^3 - 5x^2 + 2x - 7. What is the value of x at which the function has a local minimum?
A.
1
B.
2
C.
0
D.
1.5
Show Answer & Explanation
Correct Answer: A
To find local minima, we take the derivative f'(x) = 9x^2 - 10x + 2 and set it to zero. Solving gives x = 1 as a critical point. Testing values around it shows it's a local minimum.
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Applications of Differentiation — ATAR (Australia) ATAR Mathematics Methods Practice Questions Online
This page contains 149 practice MCQs for the chapter Applications of Differentiation in ATAR (Australia) ATAR Mathematics Methods. The questions are organized by difficulty — 41 easy, 87 medium, 21 hard — so you can choose the right level for your preparation.
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