A stationary point is where the gradient of a curve is zero, so the curve is momentarily flat. Optimisation uses that idea to find the greatest or least value of something real: an area, a volume, a cost, a height.
Check the current Cambridge Additional Mathematics 0606 syllabus for the exact content and notation in your exam year. Our Additional Mathematics learning guide shows where this module sits among the others.
What should you already know?
You need to differentiate polynomials and expressions with negative powers, which is covered in differentiation techniques. You also need to solve quadratic equations and rearrange a simple formula to make one letter the subject.
Gradients, tangents and normals from tangents, normals and rates are a natural next step, because both modules lean on the same derivative.
An orienting example
A farmer has 60 m of fencing to make three sides of a rectangular pen. The fourth side is a long wall. What dimensions give the largest area?
Step 1, name the variables: let the two sides touching the wall be x m and the side parallel to the wall be y m.
Step 2, use the constraint: the fencing gives 2x + y = 60, so y = 60 − 2x.
Step 3, build the objective function: A = xy = x(60 − 2x) = 60x − 2x².
Step 4, differentiate and set to zero: dA/dx = 60 − 4x = 0, so x = 15, and then y = 60 − 30 = 30.
Step 5, confirm and answer: d²A/dx² = −4, which is negative, so this is a maximum. A = 15 × 30 = 450 m².
The calculus took two lines. Steps 1 to 3 did the real work, and that is where most marks are won or lost.
In which order should you study it?
- Find stationary points from a derivative: solve dy/dx = 0 and find both coordinates.
- Classify stationary behaviour using an appropriate test: decide between maximum, minimum and inflection.
- Form an objective function from a constraint: turn a word problem into one variable before differentiating.
- Check endpoints as well as stationary values: handles questions that restrict the domain.
- Interpret an optimum with units and a domain: writes the answer in context and rejects values that do not fit.
Then work through the mixed practice set. A steady pace is one lesson a day, with the practice set at the weekend.
Which traps catch most students here?
- Setting y = 0 instead of dy/dx = 0, which finds where the curve crosses the axis, not where it turns.
- Stopping at the x-value when the question asks for the point or for the maximum value.
- Declaring an inflection whenever the second derivative is zero, without checking the sign change.
- Differentiating with two variables because the constraint was never used to remove one.
- Ignoring the domain, so a stationary value outside the allowed range is reported as the answer.
Each lesson shows one of these slips in full and then corrects it.
How should you use the practice set?
Attempt each question on paper before you open the answer, and write the constraint and the objective function as separate lines. The non-calculator working trainer helps with exact arithmetic, and the quadratic structure explorer is useful for seeing turning points of quadratics. The mistake log and retest queue is a good place to record which step went wrong.
When you get something wrong, read the routing notes at the end of the practice set and go back to the lesson it names. Fix the lesson, then retry a fresh question a few days later. If setting up the problem is the stubborn part, Additional Mathematics tuition is one way to work on it with a teacher.