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 37 questions on this lesson, each with an explanation.
On which interval is f(x) = 5 − x² decreasing?
Correct answer: (B) ]0, ∞[
Explanation: Step 1: f'(x) = −2x. Step 2: f is decreasing where f'(x) ≤ 0: −2x ≤ 0 ∴ x ≥ 0. ∴ f is decreasing on ]0, ∞[
Given f'(x) = 2x + 4, on which interval is f increasing?
Correct answer: (C) ]−2, ∞[
Explanation: Step 1: f is increasing where f'(x) ≥ 0. Step 2: 2x + 4 ≥ 0 ∴ x ≥ −2. ∴ f is increasing on ]−2, ∞[
Describe the monotonicity of f(x) = x³.
Correct answer: (B) increasing on all of ℝ
Explanation: Step 1: f'(x) = 3x². Step 2: 3x² ≥ 0 for every real x, and f'(x) = 0 only at the single point x = 0. Step 3: Since f'(x) never changes sign, f is increasing on all of ℝ. ∴ f is increasing on all of ℝ
Is f(x) = x² − 2x + 1 increasing or decreasing at x = 0?
Correct answer: (C) decreasing
Explanation: Step 1: f'(x) = 2x − 2. Step 2: f'(0) = 2(0) − 2 = −2. Step 3: f'(0) = −2 is negative, so f is decreasing at x = 0. ∴ Decreasing
Which of the following describes the function f(x) = (2x + 1)/(x − 3)?
Correct answer: (C) decreasing on ]−∞, 3[ and on ]3, ∞[
Explanation: Step 1: The domain of f is ℝ − {3}. Step 2: f′(x) = [2(x − 3) − (2x + 1)]/(x − 3)² = −7/(x − 3)². Step 3: f′(x) is negative for every x ≠ 3. Step 4: f is decreasing on ]−∞, 3[ and on ]3, ∞[ (not on ℝ, since 3 is not in the domain).
On which set of intervals is f(x) = 2x³ − 9x² + 12x increasing?
Correct answer: (D) ]−∞, 1[ and ]2, ∞[
Explanation: Step 1: f'(x) = 6x² − 18x + 12 = 6(x − 1)(x − 2). Step 2: The roots are x = 1 and x = 2. Step 3: f'(x) is positive for x < 1 and for x > 2. ∴ f is increasing on ]−∞, 1[ and ]2, ∞[
Determine the increasing and decreasing intervals of f(x) = x² − 8 ln x.
Correct answer: (C) decreasing on ]0, 2[, increasing on ]2, ∞[
Explanation: Step 1: The domain of f is ]0, ∞[. Step 2: f′(x) = 2x − 8/x = 2(x² − 4)/x. Step 3: In the domain f′(x) = 0 only at x = 2; f′ is negative on ]0, 2[ and positive on ]2, ∞[. Step 4: f is decreasing on ]0, 2[ and increasing on ]2, ∞[
For which values of k is the function f(x) = kx − sin x increasing on ℝ?
Correct answer: (C) k ≥ 1
Explanation: Step 1: f'(x) = k − cos x, and cos x takes every value in [−1, 1]. Step 2: f'(x) ≥ 0 for all x ⇔ k ≥ max(cos x) = 1 (at k = 1, f' vanishes only at isolated points, so f is still increasing). ∴ k ≥ 1
The function f(x) = x³ + ax² + bx is decreasing on ]−1, 3[ and increasing on ]−∞, −1[ and on ]3, ∞[. Find a + b.
Correct answer: (D) −12
Explanation: Step 1: The sign of f′ changes at x = −1 and x = 3, so f′(−1) = 0 and f′(3) = 0. Step 2: f′(x) = 3x² + 2ax + b, so f′(x) = 3(x + 1)(x − 3) = 3x² − 6x − 9. Step 3: Comparing: 2a = −6 gives a = −3, and b = −9. Step 4: a + b = −12.
For which values of a is f(x) = x³ + ax² + 3x increasing for all real x?
Correct answer: (C) −3 ≤ a ≤ 3
Explanation: Step 1: f'(x) = 3x² + 2ax + 3. Step 2: f is increasing for all x when f'(x) ≥ 0 for all x, i.e. the discriminant ≤ 0. Step 3: Discriminant = (2a)² − 4(3)(3) = 4a² − 36 ≤ 0 ∴ a² ≤ 9. Step 4: a² ≤ 9 ∴ −3 ≤ a ≤ 3. ∴ −3 ≤ a ≤ 3
The first 10 questions in every subject are free, each with an explanation.