Applications of derivatives · Definition

Local maxima and minima

calculusapplications

Before you start you need to know

Local extrema are the peaks and valleys of a function near each point. To find them, locate the critical points and classify them using derivatives.

Definition

A local maximum of \(f\) at \(c\) is a value \(f(c)\) greater than or equal to the values of \(f\) on some neighborhood of \(c\); a local minimum, less than or equal.

\[f'(c)=0,\ f''(c)>0\ \Rightarrow\ \text{local minimum}\]

A local maximum is a point higher than all its nearby neighbors; a local minimum, lower. They need not be the highest or lowest of the whole function.

For \(f(x)=x^{3}-3x\), the derivative \(f'(x)=3x^{2}-3\) vanishes at \(x=\pm1\). Since \(f''(x)=6x\) equals \(-6\) at \(x=-1\), there is a local maximum there, and at \(x=1\) it equals \(6\), giving a local minimum.

Typical mistake: assuming every point with \(f'(c)=0\) is an extremum. For \(f(x)=x^{3}\), \(f'(0)=0\), but \(f'\) does not change sign, so \(x=0\) is neither a local maximum nor minimum.

Think about it

If \(f'(a)=0\) and \(f''(a)>0\), what is at a?

A local minimum, because the graph is concave up.

Exercises 10

  1. At which value of x does \(f(x)=x^{3} - 3 x\) have a local maximum?
  2. At which value of x does \(f(x)=2 x^{3} - 9 x^{2} + 12 x\) have a local maximum?
  3. At which value of x does \(f(x)=x^{3} - 12 x + 5\) have a local maximum?
  4. At which value of x does \(f(x)=- x^{2} + 6 x\) have a local maximum?
  5. When is \(c\) a critical point of \(f\)?
  6. What does it mean for \(f\) to have a local maximum at \(c\)?
  7. What kind of extremum does \(f(x)=x^{2}-4x+1\) have at \(x=2\)?
  8. Is it true that if \(f'(c)=0\), then \(c\) is a local extremum of \(f\)?
  9. Which test classifies a critical point by the sign of \(f''\)?
  10. At which value of x does \(f(x)=x^{4}-2x^{2}\) have a local maximum?

What comes next

References

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