Applications of derivatives · Definition

Concavity and inflection points

calculusapplicationsgraphs

Before you start you need to know

The second derivative shows how a graph bends, that is, whether the slope increases or decreases. It classifies concavity and locates inflection points.

Definition

If \(f^{\prime\prime}>0\) on an interval, the graph is concave up; if \(f^{\prime\prime}<0\), concave down. An inflection point is where the concavity changes, and there \(f^{\prime\prime}(c)=0\) or does not exist.

\[\begin{gathered}f''>0\ \Rightarrow\ \text{concave up}\\ f''<0\ \Rightarrow\ \text{concave down}\end{gathered}\]

The first derivative tells whether the function rises or falls; the second tells how it bends. If \(f''>0\) the graph is concave up (like a cup); if \(f''<0\), concave down.

An inflection point is where the concavity changes. There \(f''\) usually vanishes and changes sign.

Together, \(f'\) and \(f''\) let you sketch the shape of a graph without evaluating many points.

Think about it

What happens to \(f''\) at an inflection point?

It changes sign (it is usually 0 there).

Exercises 10

  1. At which value of x does \(f(x)=x^{3} - 6 x^{2} + 9 x\) have its inflection point?
  2. At which value of x does \(f(x)=x^{3} - 3 x^{2} + 1\) have its inflection point?
  3. At which values of x does \(f(x)=x^{4} - 6 x^{2}\) have inflection points?
  4. For \(f(x)=x^{4}\), \(f''(0)=0\). Does f have an inflection point at x = 0?
  5. What does \(f''(x)>0\) on an interval indicate?
  6. What is an inflection point of a graph?
  7. On which interval is the graph of \(f(x)=x^{3}\) concave up?
  8. Where is the inflection point of \(f(x)=x^{3}-3x\)?
  9. At which values of x does \(f(x)=x^{4}-4x^{3}\) have inflection points?
  10. What must happen to \(f''\) at an inflection point?

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.