Stationary Points and Their Nature
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Question
What is a stationary point?
Answer
A point where f'(x) = 0. The tangent is horizontal — the function momentarily stops increasing or decreasing.
💡 Hint
f'(x)=0 → flat tangent.
Question
What is a local MAXIMUM?
Answer
A stationary point where the function changes from INCREASING to DECREASING. f'(x) goes from + to −. The point is the highest nearby.
💡 Hint
Peak: + before, − after.
Question
What is a local MINIMUM?
Answer
A stationary point where the function changes from DECREASING to INCREASING. f'(x) goes from − to +. The point is the lowest nearby.
💡 Hint
Valley: − before, + after.
Question
How do you find and classify stationary points?
Answer
1) Find f'(x). 2) Solve f'(x) = 0. 3) Use a sign diagram: if + then − → local max; if − then + → local min. 4) Find the y-value using f(x).
💡 Hint
Sign diagram to classify: look either side of critical x.
Question
f(x) = x³ − 3x. Find and classify the stationary points.
Answer
f'(x) = 3x² − 3 = 3(x−1)(x+1). Critical points: x = 1 and x = −1. Sign: +,−,+ → x=−1 local max, x=1 local min. y values: f(−1)=2, f(1)=−2.
💡 Hint
Factor f'(x) to find critical x, then sign diagram.
Question
What is a point of inflection? Is it a stationary point?
Answer
An inflection point is where concavity changes. It is only a stationary point if f'(x) = 0 there too (a "saddle point" like x=0 on y=x³).
💡 Hint
Inflection ≠ stationary by itself.
Question
f(x) = 2x³ − 3x². Find the local maximum point.
Answer
f'(x) = 6x² − 6x = 6x(x−1). Critical x: 0 and 1. Sign diagram: + before x=0, − between 0 and 1. So x=0 is local max. f(0) = 0.
💡 Hint
Check sign BOTH sides of each critical point.
Question
What does the second derivative test say? (f''(x) method)
Answer
At a critical point where f'(x)=0: if f''(x) < 0 → local max; if f''(x) > 0 → local min; if f''(x) = 0 → inconclusive, use sign diagram.
💡 Hint
Second derivative shortcut — but sign diagram always works.
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Topic 5.6 hub
Stationary points, local maximum and minimum
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