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All 9 Flashcards — Stationary points
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Question
What is a stationary point?
Answer
A point where the gradient is zero: f'(x) = 0.
Question
How do you find stationary points?
Answer
Differentiate and solve f'(x) = 0.
Question
How do you classify a stationary point with the second derivative?
Answer
f''(x) > 0 → minimum, f''(x) < 0 → maximum.
Question
What if f''(x) = 0 at a stationary point?
Answer
The test is inconclusive; check the sign of f'(x) on each side.
Question
How do you find the y-coordinate of a stationary point?
Answer
Substitute the x-value into the original function f(x).
Question
How many stationary points does a cubic usually have?
Answer
Two — a local maximum and a local minimum (or none).
Question
What does the first-derivative (sign) test do?
Answer
Checks the sign of f' just before and after: +→− max, −→+ min.
Question
Stationary points of x³ − 6x² + 9x?
Answer
x = 1 and x = 3 (from 3(x−1)(x−3) = 0).
Question
Is a stationary point always a max or min?
Answer
No — it could be a (stationary) point of inflexion.
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Full study notes for Stationary points
Topic 5.8 hub
Optimisation & inflexion
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