# 2.16.2 Differentiation - Turning Points

#### Remind me of the rule for differentiating powers …

• If y = kxn then dy/dx = knxn-1

#### What is a turning point?

• The easiest way to think of a turning point is that it is a point at which a curve changes from moving upwards to moving downwards, or vice versa
• Turning points are also called stationary points
• Ensure you are familiar with Differentiation – Basics before moving on

• At a turning point the gradient of the curve is zero.
• If a tangent is drawn at a turning point it will be a horizontal line
• Horizontal lines have a gradient of zero
• This means at a turning point the derived function (aka gradient function or derivative) equals zero

#### How do I know if a curve has turning points?

• You can see from the shape of a curve whether it has turning points or not
• At IGCSE, two types of turning point are considered:
• Maximum points – this is where the graph reaches a “peak
• Minimum points – this is where the graph reaches a “trough

• These are sometimes called local maximum/minimum points as other parts of the graph may still reach higher/lower values

#### How do I find the coordinates of a turning point?

• STEP 1  Solve the equation of the derived function (derivative) equal to zero
ie. solve dy/dx = 0
This will find the x-coordinate of the turning point
• STEP 2  To find the y-coordinate substitute the x-coordinate into the equation of the graph
ie. substitute x into “y = …

#### How do I know which point is a maximum and which is a minimum?

• There are several ways to do this
• Exam questions may ask you to “give reasons” or “justify” why you think a particular turning point is, say, a maximum
• The easiest way to do this is to recognise the shape of the curve
• … either from a given sketch of the curve
• … a sketch of the curve you can quickly draw yourself
(You may even be asked to do this as part of a question)
• … the equation of the curve
• For parabolas (quadratics) it should be obvious …
• … a positive parabola (positive x2 term) has a minimum point
• … a negative parabola (negative x2 term) has a maximum point

• Cubic graphs are also easily recognisable …
• … a positive cubic has a maximum point on the left, minimum on the right
• … a negative cubic has a minimum on the left, maximum on the right

#### Exam Tip

Remember to read the questions carefully – sometimes only the x-coordinate of a turning point is required.

Differentiating accurately is crucial in leading to equations you can work with and solve.

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