Stationary points and optimisation · Grade 12

In Grade 12 you learn: stationary points.

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.
• 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.