Optimisation & inflexion
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What is a stationary point?
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All Flashcards in Topic 5.8
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5.8.19 cards
What is a stationary point?
A point where the gradient is zero: f'(x) = 0.
How do you find stationary points?
Differentiate and solve f'(x) = 0.
How do you classify a stationary point with the second derivative?
f''(x) > 0 → minimum, f''(x) < 0 → maximum.
What if f''(x) = 0 at a stationary point?
The test is inconclusive; check the sign of f'(x) on each side.
How do you find the y-coordinate of a stationary point?
Substitute the x-value into the original function f(x).
How many stationary points does a cubic usually have?
Two — a local maximum and a local minimum (or none).
What does the first-derivative (sign) test do?
Checks the sign of f' just before and after: +→− max, −→+ min.
Stationary points of x³ − 6x² + 9x?
x = 1 and x = 3 (from 3(x−1)(x−3) = 0).
Is a stationary point always a max or min?
No — it could be a (stationary) point of inflexion.
5.8.29 cards
What are the steps of an optimisation problem?
Model the quantity → use the constraint to get one variable → differentiate → solve f'(x)=0 → classify → answer.
Why must the quantity be in one variable?
You can only differentiate a function of a single variable.
How do you eliminate the second variable?
Use the given constraint (fixed perimeter, total length, etc.) to substitute.
How do you find the optimal value?
Solve f'(x) = 0.
How do you confirm a maximum?
Show f''(x) < 0 (or that f' changes + to −) at that x.
What form do many cost problems take?
A reciprocal model like T = ax + b/x.
How do you differentiate b/x?
Write it as bx⁻¹; its derivative is −bx⁻² = −b/x².
Why keep only positive solutions?
Lengths, volumes and similar physical quantities can't be negative.
What should the final answer give?
Whatever the question asks — dimensions and/or the maximum/minimum value.
5.8.39 cards
What is a point of inflexion?
A point where the curve changes concavity (concave up ↔ concave down).
What two conditions define a point of inflexion?
f''(x) = 0 AND f'' changes sign through that point.
How do you find a point of inflexion?
Solve f''(x) = 0, confirm f'' changes sign, then find y from f(x).
Is f''(x) = 0 enough for an inflexion?
No — f'' must also change sign; e.g. y = x⁴ at x = 0 is not an inflexion.
How do you confirm the sign change?
Test f'' at a value just below and just above the candidate x.
Where do you get the y-coordinate?
From the original function f(x).
Point of inflexion of y = x³?
(0, 0).
Why is y = x⁴ not inflexion at 0?
f''(x) = 12x² ≥ 0 on both sides — no sign change.
Concavity each side of an inflexion?
Opposite: one side concave up, the other concave down.
Topic 5.8 study notes
Full notes & explanations for Optimisation & inflexion
Math AA SL exam skills
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