14.6 Stationary Points

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📖 Main Content

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1. What Is a Stationary Point?

Imagine you are walking along a hilly path. Sometimes you reach the very top of a hill — you can't go any higher. Sometimes you reach the very bottom of a dip — you can't go any lower. At those exact moments, you are neither going up nor going down. You are standing still, in terms of the slope.

This is exactly what a stationary point is on a curve.

Recall that the gradient of a curve at any point is given by dy/dx (the first derivative). So:

$$\frac{dy}{dx} = 0 \quad \text{at every stationary point}$$

There are two types of stationary points you need to know:

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2. Maximum Points

A maximum point is a stationary point where the curve reaches a local peak — the y-value at this point is higher than the y-values of all the nearby points around it.

Think of it as the very top of a hill.

What happens to the gradient around a maximum point?

• Before the maximum (to the left): the curve is going up, so the gradient is positive (+)
• At the maximum: the curve is flat, so the gradient is zero (0)
• After the maximum (to the right): the curve is going down, so the gradient is negative (−)

So the gradient changes from positive → zero → negative at a maximum point.

Gradient pattern at a maximum:   +   →   0   →   −
Curve shape:                     ↗       —       ↘

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3. Minimum Points

A minimum point is a stationary point where the curve reaches a local dip — the y-value at this point is lower than the y-values of all the nearby points around it.

Think of it as the very bottom of a valley.

What happens to the gradient around a minimum point?

• Before the minimum (to the left): the curve is going down, so the gradient is negative (−)
• At the minimum: the curve is flat, so the gradient is zero (0)
• After the minimum (to the right): the curve is going up, so the gradient is positive (+)

So the gradient changes from negative → zero → positive at a minimum point.

Gradient pattern at a minimum:   −   →   0   →   +
Curve shape:                     ↘       —       ↗