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Calculus foundations
Lesson 10 of 75
Review

Optimisation and curve analysis

Objective. Classify stationary points and solve constrained models.

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Learn: the key idea

Optimisation uses derivatives to locate stationary points (where f'(x)=0) and classify them as local maxima, minima, or saddle points using the first or second derivative test, then analyses constraints to find global extrema on a given domain. The second derivative test says if f''(x₀)>0 the point is a local minimum, if f''(x₀)<0 it is a local maximum, and if f''(x₀)=0 the test is inconclusive. For f(x)=x³−3x, f'(x)=3x²−3=0 gives x=±1; f''(x)=6x, so f''(1)=6>0 (minimum) and f''(−1)=−6<0 (maximum). Constrained models often substitute a relationship between variables to reduce to a single-variable function before differentiating.

Worked example

Find and classify the stationary points of f(x)=x³−3x using optimisation techniques.

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Illustrated card showing 3
  1. 1Differentiate: f'(x)=3x²−3, and set f'(x)=0 to get x²=1, so x=1 or x=−1.
  2. 2Compute the second derivative: f''(x)=6x.
  3. 3Evaluate f''(1)=6>0, so x=1 is a local minimum; f''(−1)=−6<0, so x=−1 is a local maximum.
  4. 4Check values: f(1)=1−3=−2, f(−1)=−1+3=2, confirming distinct extrema.

Answer: Local maximum at x=−1 (value 2); local minimum at x=1 (value −2)

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Common mistake. Students find f'(x)=0 and declare every stationary point a maximum or minimum without checking the second derivative or sign change, missing inflection points such as f(x)=x³ at x=0, which is neither.

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