Pure Mathematics — Second: Differential & Integral Calculus · Mathematics. These are 10 questions from the Magmo3 bank on this lesson, from easy to hard. Solve each one, then open "Show answer and explanation". Magmo3 has 67 questions on this lesson, each with an explanation.
What is the maximum value of the function f(x) = x − x² ?
Correct answer: (B) 1/4
Explanation: Step 1: f(x) = x − x² Step 2: Differentiate: f'(x) = 1 − 2x Step 3: Set f'(x) = 0: 1 − 2x = 0, so x = 1/2 Step 4: f''(x) = −2 less than 0, so x = 1/2 gives a maximum Step 5: f(1/2) = 1/2 − (1/2)² = 1/2 − 1/4 = 1/4 ∴ Maximum value = 1/4
Given f'(x) = x − 4, at which value of x does f have a critical point?
Correct answer: (C) x = 4
Explanation: Step 1: Critical points occur where f'(x) = 0. Step 2: x − 4 = 0 ∴ x = 4. ∴ Critical point x = 4
For the function f(x) = x² − 6x + 5, find the critical point (where f'(x) = 0).
Correct answer: (C) x = 3
Explanation: Step 1: A critical point occurs where f'(x) = 0. Step 2: f'(x) = 2x − 6. Step 3: 2x − 6 = 0 ∴ x = 3. ∴ The critical point is x = 3
For f(x) = x³ − 12x + 1, what is the nature of the critical point at x = −2?
Correct answer: (A) local maximum
Explanation: Step 1: f'(x) = 3x² − 12 = 0 ∴ x² = 4 ∴ x = ±2. Step 2: f''(x) = 6x. Step 3: f''(−2) = 6(−2) = −12 is negative, so x = −2 is a local maximum. ∴ Local maximum
Find the local minimum value of f(x) = x³ − 3x² + 4.
Correct answer: (D) 0
Explanation: Step 1: f'(x) = 3x(x − 2); at x = 2, f'(x) changes from negative to positive, so x = 2 is a local minimum. Step 2: f(2) = (2)³ − 3(2)² + 4 = 8 − 12 + 4 = 0. ∴ Local minimum value = 0
Given f'(x) = (x + 1)²(x − 3), what is the nature of x = −1?
Correct answer: (A) neither maximum nor minimum
Explanation: Step 1: Critical points from f'(x) = 0: x = −1 (double) and x = 3. Step 2: Near x = −1, (x + 1)² ≥ 0 and (x − 3) stays negative, so f'(x) keeps the same sign on both sides. Step 3: Since f'(x) does not change sign at x = −1, it is neither a maximum nor a minimum. ∴ Neither maximum nor minimum
For x greater than 0, what is the minimum value of f(x) = x + 9/x ?
Correct answer: (D) 6
Explanation: Step 1: f(x) = x + 9/x = x + 9x⁻¹ Step 2: Differentiate: f'(x) = 1 − 9/x² Step 3: Set f'(x) = 0: 1 = 9/x², so x² = 9, x = 3 (taking x greater than 0) Step 4: f''(x) = 18/x³ greater than 0 at x = 3, so it is a minimum Step 5: f(3) = 3 + 9/3 = 3 + 3 = 6 ∴ Minimum value = 6
The curve of f(x) = ax³ + bx² has a local extremum at the point (1, 2). Which of the following is correct?
Correct answer: (B) a = −4, b = 6, local maximum
Explanation: Step 1: f'(1) = 3a + 2b = 0 and f(1) = a + b = 2 ⇒ a = −4, b = 6. Step 2: f''(x) = 6ax + 2b ⇒ f''(1) = −24 + 12 = −12 < 0, so (1, 2) is a local maximum. ∴ a = −4, b = 6, local maximum
For which values of k does the equation x³ − 3x + k = 0 have three distinct real roots?
Correct answer: (D) −2 < k < 2
Explanation: Step 1: f(x) = x³ − 3x + k has f'(x) = 3(x − 1)(x + 1): local maximum f(−1) = 2 + k, local minimum f(1) = k − 2. Step 2: The curve cuts the x-axis three times ⇔ local max > 0 and local min < 0 ⇔ k > −2 and k < 2. ∴ −2 < k < 2
The function f(x) = x³ + ax² + bx has local extrema at x = 1 and x = 3. What is its local maximum value?
Correct answer: (D) 4
Explanation: Step 1: f'(x) = 3x² + 2ax + b has roots 1 and 3: 3 + 2a + b = 0 and 27 + 6a + b = 0 ⇒ a = −6, b = 9. Step 2: f'(x) = 3(x − 1)(x − 3) changes from + to − at x = 1, so the local maximum value = f(1) = 1 − 6 + 9 = 4 (f(3) = 0 is the local minimum). ∴ 4
The first 10 questions in every subject are free, each with an explanation.