Stationary points and optimisation · Year 13
In Year 13 you learn: optimisation problems.
Once you can find gradients, you can find where a curve turns, where it rises or falls, and the equations of its tangents and normals. This makes differentiation a powerful tool for optimisation: finding the biggest area, the cheapest design or the shortest distance.
Key facts
- Stationary points: solve dy/dx = 0.
- (d^2y)/(dx^2) > 0 gives a minimum and < 0 gives a maximum. If it is 0, check the sign of dy/dx on either side.
- f is increasing where f′(x) > 0 and decreasing where f′(x) < 0.
- A point of inflection is where the curve changes concavity. There f″(x) changes sign.
- Tangent: y − y₁ = m(x − x₁). The normal has gradient −1/m.
- Optimisation: use the constraint to write the quantity in terms of ONE variable. Differentiate and set it to 0. Check it is the max or min you want.
Worked examples
Example 1
Find and classify the stationary points of y = x^3 − 3x^2 − 9x + 5.
Answer: Maximum (−1, 10); minimum (3, −22)
Example 2
An open box with a square base must hold 32,000 cm³. Find the dimensions that use the least card.
Answer: base 40 cm × 40 cm, height 20 cm (4,800 cm² of card)