Stationary Points & Turning Points (CIE A Level Maths: Pure 1)

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Stationary Points & Turning Points

What are stationary points?

  • A stationary point is any point on a curve where the gradient is zero
  • To find stationary points of a function f(x)

Step 1: Find the first derivative f'(x)

Step 2: Solve f'(x) = 0 to find the x-coordinates of the stationary points

Step 3: Substitute those x-coordinates into f(x) to find the corresponding y-coordinates

  • A stationary point may be either a local minimum, a local maximum, or a point of inflection

 

Stat Point Illustr 1, A Level & AS Maths: Pure revision notes

Stationary points on quadratics

  • The graph of a quadratic function (ie a parabola) only has a single stationary point
  • For an 'up' parabola this is the minimum; for a 'down' parabola it is the maximum (no need to talk about 'local' here)

Stationary Points min max for parabola illustr, A Level & AS Maths: Pure revision notes

  • The y value of the stationary point is thus the minimum or maximum value of the quadratic function
  • For quadratics especially minimum and maximum points are often referred to as turning points

How do I determine the nature of stationary points on other curves?

  • For a function f(x) there are two ways to determine the nature of its stationary points 
  • Method A: Compare the signs of the first derivative (positive or negative) a little bit to either side of the stationary point
    • (After completing Steps 1 - 3 above to find the stationary points)

Step 4: For each stationary point find the values of the first derivative a little bit 'to the left' (ie slightly smaller x value) and a little bit 'to the right' (slightly larger x value) of the stationary point

Stat Points left right proviso, A Level & AS Maths: Pure revision notes

Step 5: Compare the signs (positive or negative) of the derivatives on the left and right of the stationary point

    • If the derivatives are negative on the left and positive on the right, the point is a local minimum
    • If the derivatives are positive on the left and negative on the right, the point is a local maximum
    • If the signs of the derivatives are the same on both sides (both positive or both negative) then the point is a point of inflection

incr decr min max, A Level & AS Maths: Pure revision notes Stationary Points point of inflection, A Level & AS Maths: Pure revision notes Stationary Points Method A flow chart, A Level & AS Maths: Pure revision notes

Method B: Look at the sign of the second derivative (positive or negative) at the stationary point

(After completing Steps 1 - 3 above to find the stationary points)

Step 4: Find the second derivative f''(x)

Step 5: For each stationary point find the value of f''(x) at the stationary point (ie substitute the x-coordinate of the stationary point into f''(x) )

    • If f''(x) is positive then the point is a local minimum
    • If f''(x) is negative then the point is a local maximum
    • If f''(x) is zero then the point could be a local minimum, a local maximum OR a point of inflection (use Method A to determine which)

Stationary Points Method B flow chart, A Level & AS Maths: Pure revision notes

Examiner Tip

  • Usually using the second derivative (Method B above) is a much quicker way of determining the nature of a stationary point.
  • However, if the second derivative is zero it tells you nothing about the point.
    • In that case you will have to use Method A (which always works – see the Worked Example).

Worked example

Stat Point Example_1, A Level & AS Maths: Pure revision notes

7-2-4-stat-point-example-2-fixed-with-watermark

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Roger

Author: Roger

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.