Radford Mathematics IB Mathematics resources • AA HL

IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus

Second Derivative and Graphical Behaviour

Use the second derivative to describe concavity and identify changes of curvature.

AA HL · SL 5.7

Learning goal

Interpret the second derivative, concavity and points of inflexion.

Syllabus link

AA HL: SL 5.7 · Topic 5 Calculus.

Big idea

The first derivative describes slope; the second derivative describes how that slope is changing.

Key relationship

\(f\'\'(x)>0\Rightarrow\text{concave up}\)

1

Core idea

The first derivative describes slope; the second derivative describes how that slope is changing.

Interpret the second derivative, concavity and points of inflexion.

2

Key results and method

\[f\'\'(x)>0\Rightarrow\text{concave up}\]
\[f\'\'(x)<0\Rightarrow\text{concave down}\]
\[\text{POI requires a change in concavity}\]
  1. Find \(f''(x)\).
  2. Locate candidates where \(f''(x)=0\) or is undefined.
  3. Check the sign of \(f''\) on both sides.
  4. Find the corresponding point on the original graph.
3

Worked example

Question

Find the point of inflexion of \(f(x)=x^3-6x^2+2\).

Solution
  1. \(f\'(x)=3x^2-12x\), so \(f\'\'(x)=6x-12\).
  2. \(f\'\'(x)=0\Rightarrow x=2\).
  3. The second derivative changes from negative to positive at 2.
Answer: POI at \((2,f(2))=(2,-14)\).
4

Practice

Try these questions before moving on.

  1. Find concavity intervals for a quartic.
  2. Give an example showing that \(f''(a)=0\) alone is insufficient for an inflexion point.
  3. Match a graph of \(f\) to a graph of \(f''\).

Search the worksheet library for more practice →

5

Tutorials and premium resources

▶ TI-Nspire: first and second derivative functions ▶ TI-Nspire: points of inflection
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