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CIE A-Level Maths Study Notes

1.7.4 Stationary Points

Differentiation is essential for understanding how functions behave. In this section, the focus is on stationary points, which are crucial in analysing the nature of functions.

Stationary Points, Increasing and Decreasing Functions

  • Increasing Functions: When f'(x) > 0, the function is increasing.
  • Decreasing Functions: When f'(x) < 0, the function is decreasing.
  • Stationary Points: When f(x)=0f'(x) = 0, we encounter a stationary point.

Example: Analysing Function Behaviour

For the function y=x3+3x29x+4y = x^3 + 3x^2 - 9x + 4, determine the values of xx where the graph is increasing or decreasing.

Solution:

1. Finding the Derivative: dydx=3x2+6x9\frac{dy}{dx} = 3x^2 + 6x - 9

2. Identifying Stationary Points:

Solve 3x2+6x9=03x^2 + 6x - 9 = 0 to find x=1x = 1 and x=3x = -3.

3. Analysing Intervals:

  • The derivative forms a U-shaped parabola.
  • The function is increasing for x < -3 and x > 1.
  • The function is decreasing for -3 < x < 1
stationary points illustration

Determining the Nature of Stationary Points

1. Find the Second Derivative:

Compute d2ydx2\frac{d^2y}{dx^2}.

2. Substitute the Stationary Point:

Substitute the xx-value of the stationary point into the second derivative.

3. Interpret the Result:

  • If \frac{d^2y}{dx^2} > 0, it's a minimum point.
  • If \frac{d^2y}{dx^2} < 0, it's a maximum point.

Example: Identifying Stationary Points

Find the stationary points for y=x3+3x2y = x^3 + 3x^2 and determine their nature.

Solution:

1. First Derivative: dydx=3x2+6x\frac{dy}{dx} = 3x^2 + 6x

2. Stationary Points:

Set 3x2+6x=03x^2 + 6x = 0 to find x=2x = -2 and x=0x = 0 .

3. Second Derivative: d2ydx2=6x+6\frac{d^2y}{dx^2} = 6x + 6

4. Nature of Stationary Points:

  • For x=2x = -2, d2ydx2=6\frac{d^2y}{dx^2} = -6 (Maximum Point at (2,4)(-2, 4)).
  • For x=0x = 0, d2ydx2=6\frac{d^2y}{dx^2} = 6 (Minimum Point at (0,0)(0, 0)).
Stationary points illustration
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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