The second derivative & concavity
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Flip to reveal answersWhat is the second derivative f''(x)?
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Question
What is the second derivative f''(x)?
Answer
The derivative of f'(x) — the rate of change of the gradient. Written f''(x) or d²y/dx².
Question
What does f''(x) > 0 tell you about a curve?
Answer
It is concave up (valley shaped) there — the gradient is increasing.
Question
What does f''(x) < 0 tell you about a curve?
Answer
It is concave down (dome shaped) there — the gradient is decreasing.
Question
How do you find a point of inflexion?
Answer
Solve f''(x) = 0, then confirm f'' changes sign across that x (concavity flips).
Question
State the second-derivative test for a stationary point.
Answer
At f'(a) = 0: f''(a) < 0 → local maximum; f''(a) > 0 → local minimum; f''(a) = 0 → inconclusive.
Question
What happens if f''(a) = 0 at a stationary point?
Answer
The second-derivative test is inconclusive — go back to checking the sign of f' on each side.
Question
In motion, what is the second derivative of displacement?
Answer
Acceleration (displacement → velocity → acceleration by differentiating twice).
Question
Find f''(x) for f(x) = x³ − 6x² + 5x + 2.
Answer
f'(x) = 3x² − 12x + 5, so f''(x) = 6x − 12.
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Topic 5.10 hub
Second derivatives (HL only)
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