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 44 questions on this lesson, each with an explanation.
Find the inflection point of the curve f(x) = x³ − 3x² + 2.
Correct answer: (A) (1, 0)
Explanation: Step 1: First derivative: f'(x) = 3x² − 6x Step 2: Second derivative: f''(x) = 6x − 6 Step 3: Set f''(x) = 0: 6x − 6 = 0, so x = 1 Step 4: f''(x) changes sign from negative (x less than 1) to positive (x greater than 1), so an inflection point exists Step 5: y-value: f(1) = 1 − 3 + 2 = 0 ∴ The inflection point is (1, 0)
Using the second derivative test, classify the critical point of f(x) = x² − 10x + 1.
Correct answer: (B) local minimum
Explanation: Step 1: f'(x) = 2x − 10 = 0 ∴ x = 5. Step 2: f''(x) = 2. Step 3: f''(5) = 2 is positive, so by the second derivative test x = 5 is a local minimum. ∴ Local minimum
At which value of x does the curve f(x) = x³ − 6x² + 5x have an inflection point?
Correct answer: (C) x = 2
Explanation: Step 1: f'(x) = 3x² − 12x + 5 Step 2: f''(x) = 6x − 12 Step 3: Set f''(x) = 0: 6x − 12 = 0, so x = 2 Step 4: f''(x) changes sign from negative to positive at x = 2, so it is an inflection point ∴ x = 2
On which interval is the curve f(x) = −x³ + 3x² convex upward?
Correct answer: (C) ]1, ∞[
Explanation: Step 1: f'(x) = −3x² + 6x Step 2: f''(x) = −6x + 6 Step 3: The curve is convex upward where f''(x) less than 0: −6x + 6 less than 0, so −6x less than −6, giving x greater than 1 ∴ The curve is convex upward on ]1, ∞[
For the function f where f(x) = 2x³ + 3x² − 12x, on the interval ]−2, −1/2[ the curve of f is ..........
Correct answer: (A) decreasing and convex upward
Explanation: Step 1: f′(x) = 6x² + 6x − 12 = 6(x + 2)(x − 1) < 0 on ]−2, 1[, so f is decreasing on ]−2, −1/2[. Step 2: f″(x) = 12x + 6 < 0 when x < −1/2, so the curve is convex upward there. ∴ decreasing and convex upward (f″ < 0 means convex upward, f″ > 0 means convex downward).
Find the measure of the angle that the tangent at the inflection point of the curve f(x) = x³ − 6x² + 11x − 6 makes with the positive direction of the x-axis.
Correct answer: (B) 3π/4
Explanation: Step 1: f′(x) = 3x² − 12x + 11 and f″(x) = 6x − 12. Step 2: f″(x) = 0 at x = 2 and f″ changes sign there, so the inflection point is at x = 2. Step 3: Slope of the tangent there: f′(2) = 12 − 24 + 11 = −1. Step 4: tan θ = −1 with θ between 0 and π, so θ = 3π/4. Step 5: The angle is 3π/4.
Use the second derivative test to find the local maximum value of f(x) = x³ − 3x² − 24x + 5.
Correct answer: (C) 33
Explanation: Step 1: f′(x) = 3x² − 6x − 24 = 3(x − 4)(x + 2), so f′(x) = 0 at x = −2 and x = 4. Step 2: f″(x) = 6x − 6. Step 3: f″(−2) = −18, which is negative, so f(−2) is a local maximum value; f″(4) = 18 is positive, so f(4) is a local minimum value. Step 4: f(−2) = −8 − 12 + 48 + 5 = 33. Step 5: The local maximum value is 33.
The curve of f(x) = x³ + ax² + bx has an inflection point at (1, −1). Find a + b.
Correct answer: (C) −2
Explanation: Step 1: f''(x) = 6x + 2a = 0 at x = 1 ⇒ a = −3. Step 2: f(1) = 1 − 3 + b = −1 ⇒ b = 1, so a + b = −2. ∴ −2
The increasing, decreasing and convexity table of a function f shows: f′(x) > 0 on ]−∞, −1[ and on ]1, ∞[, f′(x) < 0 on ]−1, 1[, f″(x) < 0 on ]−∞, 0[, f″(x) > 0 on ]0, ∞[, a local maximum value 3 and a local minimum value −1. Then f(x) = ..........
Correct answer: (A) x³ − 3x + 1
Explanation: Step 1: f′ changes from + to − at x = −1 and from − to + at x = 1, so the critical points are ±1; x³ + 3x + 1 has f′ = 3x² + 3 > 0 and no critical points. Step 2: x³ − 3x + 1: f′ = 3x² − 3 (+, −, +) and f″ = 6x (− then +); f(−1) = 3 and f(1) = −1 ✓. Step 3: 3x − x³ + 1 has the opposite signs of f′ (local minimum at −1), and x³ − 3x − 1 gives local values 1 and −3. ∴ f(x) = x³ − 3x + 1
f is a polynomial function of the third degree whose increasing, decreasing table shows: f′(x) > 0 on ]−∞, −2[, f′(x) < 0 on ]−2, 2[, f′(x) > 0 on ]2, ∞[, with f(−2) = 3 and f(2) = −1. How many points does the curve of f have in common with the x-axis?
Correct answer: (D) three
Explanation: Step 1: f(−2) = 3 is a local maximum value and f(2) = −1 is a local minimum value. Step 2: On ]−∞, −2[ f increases (from −∞, as for any such cubic) up to 3, so it crosses the x-axis once; on ]−2, 2[ it decreases from 3 to −1, crossing once more. Step 3: On ]2, ∞[ it increases from −1 without bound, crossing a third time. ∴ three points (the curve would meet the axis in only one point if both local values had the same sign).
The first 10 questions in every subject are free, each with an explanation.