The Second Derivative, Concavity, and Points of Inflection

Expert reviewed • 08 January 2025 • 7 minute read

HSC Maths Advanced Syllabus

  • define and interpret the concept of the second derivative as the rate of change of the first derivative function in a variety of contexts, for example recognise acceleration as the second derivative of displacement with respect to time
    • understand the concepts of concavity and points of inflection and their relationship with the second derivative
    • use the second derivative to determine concavity and the nature of stationary points
    • understand that when the second derivative is equal to 0 this does not necessarily represent a point of inflection

What is the use of the Second Derivative?

The second derivative is a significantly important tool in determining a function’s concavity, while also confirming the points of inflection, found using the first derivative. Similar to using the first derivative when working with graphs, we must sub in various points, while also using zero to determine the information needed.

Concavity

Concavity refers to the direction in which the graph of a function curves. It can be determined by the sign of the second derivative:

  • A function is noted as concave up, when a point substituted into the second derivative is greater than zero: f(x)>0f(x)>0 (This is known as a local or global minimum)
  • A function is noted as concave down, when a point substituted into the second derivative is less than zero: f(x)<0f(x)<0 (This is known as a local or global maximum)

Points of Inflection

As noted in the previous chapter, a point of inflection is a point on the graph of the function where the concavity changes. The point of inflections can be determined by equating the second derivative with zero.

However, as we previously learnt, a point of inflection is a stationary point, and can thus be determined by equating the first derivative of a function with zero. If there are multiple stationary points we must use the second derivative to test if any are points of inflection. This is done by substituting the x-values of the stationary points into the second derivative. As such, the ones that result in an answer of 0, are points of inflection.

Practice Question 1

The following function y=x3−3x2−3x+9y=x^3-3x^2-3x+9 has stationary points at x=1−2x=1-\sqrt{2} and x=1+2x=1+\sqrt{2}. Determine the points of inflection of the function.

To answer the question we must first find the second derivative.

y‘=3x2−6x−3y‘‘=6x−6y^`=3x^2-6x-3\\y^{``}=6x-6

Now, we must first test if any of the stationary points equate to zero when substituted into the second derivative.

y‘‘(1−2)=6(1−2)−6=−2y‘‘(1+2)=6(1+2)−6=2y^{``}(1-\sqrt{2})=6(1-\sqrt{2})-6\\=-\sqrt{2}\\y^{``}(1+\sqrt{2})=6(1+\sqrt{2})-6\\=\sqrt{2}

Because none of the stationary points equal zero, when substituted into the second derivative, they are not points of inflection.

Now, equating the second derivative with zero we find:

0=6x−66x=6x=10=6x-6\\6x=6\\x=1

Substituting into yy:

y=(1)3−3(1)2−3(1)+9=1−3−3+9=4y=(1)^3-3(1)^2-3(1)+9\\=1-3-3+9\\=4

∴\therefore This equation only has one point of inflection at (1,4).

It is important to note that using the second derivative to find the concavity of a curve is similar to using a table of signs. Both can be used, however sometimes it is more practical to use the second derivative, as it can be a quicker method.

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Stationary and Turning Points

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How to Sketch a Graph using the Derivative

Return to Module 2: Curve Sketching Using the Derivative