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All 9 Flashcards — Points of inflexion
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Question
What is a point of inflexion?
Answer
A point where the curve changes concavity (concave up ↔ concave down).
Question
What two conditions define a point of inflexion?
Answer
f''(x) = 0 AND f'' changes sign through that point.
Question
How do you find a point of inflexion?
Answer
Solve f''(x) = 0, confirm f'' changes sign, then find y from f(x).
Question
Is f''(x) = 0 enough for an inflexion?
Answer
No — f'' must also change sign; e.g. y = x⁴ at x = 0 is not an inflexion.
Question
How do you confirm the sign change?
Answer
Test f'' at a value just below and just above the candidate x.
Question
Where do you get the y-coordinate?
Answer
From the original function f(x).
Question
Point of inflexion of y = x³?
Answer
(0, 0).
Question
Why is y = x⁴ not inflexion at 0?
Answer
f''(x) = 12x² ≥ 0 on both sides — no sign change.
Question
Concavity each side of an inflexion?
Answer
Opposite: one side concave up, the other concave down.
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Full study notes for Points of inflexion
Topic 5.8 hub
Optimisation & inflexion
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