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By the end of this topic, you should be able to:
This means you will use what you know about derivatives (rates of change) to solve real-world problems — such as finding the largest possible area, the smallest amount of material needed, or the maximum volume of a container.
The word "optimize" means to make something as good as possible. In mathematics, this usually means finding the maximum (largest) or minimum (smallest) value of something in a real-life situation.
Examples of optimization problems:
All of these are solved using differentiation — the process of finding how fast something is changing.
Before diving into practical problems, let's quickly recall the key ideas you already know.
A stationary point is a point on a curve where the gradient is exactly zero. The gradient is found using the first derivative, written as dy/dx.
💡 Stationary point condition: At a stationary point, dy/dx = 0
Once you find a stationary point, you need to know whether it is a maximum or a minimum. You do this using the second derivative, written as d²y/dx².
| Second Derivative at the Point | Type of Stationary Point |
|---|---|
| d²y/dx² < 0 (negative) | Local Maximum — the curve peaks here |
| d²y/dx² > 0 (positive) | Local Minimum — the curve dips here |
🔑 Memory tip: Think of negative → frowning face → maximum (top of a hill). Think of positive → smiling face → minimum (bottom of a valley).
Every optimization problem follows the same set of steps. Learn these steps and you can solve any problem of this type.
STEP 1 — Read and draw a diagram
Read the problem carefully. Draw a picture if one is not given. Label all the measurements with letters (like x, y, r, h).
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