Classifying Stationary Points (Cambridge (CIE) IGCSE Maths: Extended): Revision Note

Classifying stationary points

What are the different types of stationary points?

• You need to know two different types of stationary points:

  • Maximum points (this is where the graph reaches a “peak”)

  • Minimum points (this is where the graph reaches a “trough”)

    • These are also called turning points

    • Deciding which is which is called classifying them (or finding the nature of them)

• If a graph has multiple turning points, the one you are interested in can be described as a local maximum or minimum point

  • Other parts of the graph may still reach higher/lower values elsewhere

How do I classify turning points using graphs?

• You can use the shape of common graphs to classify different turning points

• To classify the turning point on a quadratic curve (parabola), remember that:

  • A positive quadratic (positive x² term) must have a minimum point

  • A negative quadratic (negative x² term) must have a maximum point

• To classify two different turning points on a cubic curve, remember that:

  • A positive cubic has a maximum point on the left and a minimum on the right

  • A negative cubic has a minimum point on the left and a maximum on the right

How do I classify turning points using the gradient function (derivative)?

• An alternative method if no graph is available is to use the gradient function (derivative) $\frac{\mathrm{d}y}{\mathrm{d}x}$

  • By looking at the gradient just before and just after a turning point, you can determine if it is a maximum or a minimum

  • Find the gradient just before and just after a turning point by substituting values near to the value of x into the gradient function

  • It helps to imagine what the tangents to the curve would look like

How do I classify turning points using the second derivative?

• Another method is to use the second derivative, $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}$, which is the derivative-of-the-derivative

  • $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}$ is pronounced "d-two-y by dx-squared"

• Differentiate the equation once to get $\frac{\mathrm{d}y}{\mathrm{d}x}$  then again to get $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}$

  • This is the same as differentiating the original equation for $y$ twice

• If the stationary point is at $x=a$, substitute $x=a$ into the expression for $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}$ to get a numerical value

  • If the result is negative, $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}<0$, then the stationary point is a maximum point

    • This is often the opposite of what people expect!

  • If the result is positive,  $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}>0$, then the stationary point is a minimum point

  • If the value is zero,  $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=0$, then the test has failed

    • You will need to try another method to classify the turning point