Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Calculus — Topic 5.6
📋 Learning Intentions
- Find stationary points of a function by setting the first derivative equal to zero and solving.
- Classify stationary points as local maxima, local minima, or points of inflection using the second derivative test or a sign diagram of \( f'(x) \).
- Determine the coordinates of stationary points and state whether each is a maximum or minimum in context.
- Interpret the significance of local maxima and minima in real-world optimisation problems.
- Use the GDC to locate and verify stationary points graphically and numerically.
Real-World Introduction
A drone delivery company wants to minimise the fuel used per trip. The fuel consumption depends on flight speed: too slow wastes time and increases air resistance drag; too fast burns fuel rapidly. There is an optimal speed — a minimum point on the fuel-versus-speed curve — that the engineers must locate precisely.
Similarly, a pharmaceutical company models the concentration of a drug in the bloodstream over time. The peak concentration — the local maximum — determines whether the dose is therapeutic or toxic. Finding that maximum is not optional; it is a clinical requirement.
🌍 Why This Matters Stationary points appear wherever a quantity must be optimised: maximum profit, minimum cost, peak drug concentration, lowest energy state in physics, and highest yield in agriculture. Every optimisation problem in engineering, economics, and science reduces to finding and classifying stationary points.
Core Concepts & Definitions
The derivative \( f'(x) \) measures the gradient of a function at any point. When a curve reaches a peak or a trough, the tangent line at that point is horizontal — its gradient is zero. This observation is the foundation of the entire method.
📖 Key Definition — Stationary Point A stationary point of a function \( f(x) \) is a point where the gradient is zero: \[ f'(x) = 0 \] At a stationary point, the tangent to the curve is horizontal. The \( x \)-coordinates of stationary points are