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Stationary and Turning Points

Expert reviewed • 21 July 2024 • 7 minute read

HSC Maths Advanced Syllabus

  • use the first derivative to investigate the shape of the graph of a function
    • use the first derivative to find intervals over which a function is increasing or decreasing, and where its stationary points are located
    • use the first derivative to find intervals over which a function is increasing or decreasing, and where its stationary points are located
    • use the first derivative to investigate a stationary point of a function over a given domain, classifying it as a local maximum, local minimum or neither

Note:

Video coming soon!

What are Stationary Points?

As noted in the previous chapter stationary points are points on the graph of a function where the derivative is zero. These points are significant because they can indicate where a function reaches a maximum or minimum, or where a change in the slope or direction of the graph occurs.

Turning points on a graph, are stationary points, which curve smoothly, changing the function from decreasing to increasing, or increasing to decreasing. The different types of stationary points are as follows:

  • Local Maximum: A point where the function reaches a peak, meaning it is higher than all nearby points. At a local maximum, the derivative changes from positive to negative.
  • Local Minimum: A point where the function reaches a trough, meaning it is lower than all nearby points. At a local minimum, the derivative changes from negative to positive.
  • Point of Inflection: A point where the derivative is zero, but the function does not achieve a maximum or minimum. Instead, the function's slope changes sign around the point, or it might remain the same, indicating a change in concavity or a flat area. (We will find out how to determine points of inflection in a later chapter.)
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How to Find Stationary Points

As touched on in the previous chapter, to find any stationary points of a function, you must equate the function’s derivative with zero. By doing this, you will be able to find the turning points, points of inflection, discontinuities and any other stationary points

Once this step has been completed, it is important to construct a table of signs, to determine the gradient of the slope. This is important when it comes to graphing, as you are able to determine which stationary points are the local maximum and minimum, while making it easier to understand the overall shape of the graph. If you are not graphing, a table of values is not always necessary.

Practice Question 1

Determine the stationary points of the following function: y=x3−6x2+9x+1y=x^3-6x^2+9x+1

dydx=3x2−12x+9\frac{dy}{dx}=3x^2-12x+9

Since we have solved for the derivative, we can equate it with zero and factorise, to determine the stationary points of the function.

0=3x2−12x+9=x2−4x+3=(x−3)(x−1)x−3=0,x−1=0x=3,x=10=3x^2-12x+9\\=x^2-4x+3\\=(x-3)(x-1)\\x-3=0\quad ,\quad x-1=0\\x=3\quad , \quad x=1

Now that we have the x-values of the stationary points, we must substitute them back into the original equation to find the y-values.

y(3)=(3)3−6(3)2+9(3)+1=27−54+27+1=1y(1)=(1)3−6(1)2+9(1)+1=1−6+9+1=5y(3)=(3)^3-6(3)^2+9(3)+1\\=27-54+27+1\\=1\\y(1)=(1)^3-6(1)^2+9(1)+1\\=1-6+9+1\\=5

∴\therefore The stationary points of the function are (3,1)(3,1) and (1,5)(1,5)

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The Second Derivative, Concavity, and Points of Inflection

Return to Module 2: Curve Sketching Using the Derivative