Unit 5: Calculus
Topic 5.10: Second derivatives (HL only) Questions
Practice 12 exam-style questions for IB Math AI SL Topic 5.10. Review the question stems below, then unlock the full Question Bank to access markschemes, model answers, and AI grading.
1Find2 marks
2026
A particle moves so that its displacement is s(t) = 2t³ − 9t² + 12t metres. Find the acceleration s''(t).
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Write down what this says about the curve there.
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2026
A curve is concave down on an interval. On that interval:
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2026
The second derivative of f(x) = 4x² − 7x + 1 is:
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For f(x) = x³ − 3x² − 9x + 1, find the x-coordinate of the point of inflexion.
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2026
The growth curve of a tree is concave up then concave down. The switch point represents:
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For f(x) = x³, at x = 0 we have f'(0) = 0 and f''(0) = 0. The point (0, 0) is a:
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The temperature of a cooling cup is T(t) = + 20 °C, t minutes after pouring. Find T''(t) and state whether the cooling curve is concave up or concave down.
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Unlock Question9Classify2 marks
2026
f(x) = 2x³ − 3x² − 12x + 5 has a stationary point at x = 2. Use the second derivative to classify it.
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2026
At a stationary point of f, you find f''(a) = 0. You should:
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2026
For g(x) = x⁴ − 8x² + 3, find ALL points of inflexion (x-coordinates) and confirm each is genuine.
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2026
A bacteria population is P(t) = 50 + 30t² − t³ thousand, for 0 ≤ t ≤ 20 hours. Find when the population is growing fastest.
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