Continuity · Definition

Extreme value theorem

calculuscontinuitytheorems

Before you start you need to know

The extreme value theorem guarantees that a function continuous on a closed and bounded interval has absolute extrema. So we know they exist before we search for them.

Definition

If \(f\) is continuous on a closed and bounded interval \([a,b]\), then \(f\) attains an absolute maximum and minimum on \([a,b]\).

\[f\ \text{continuous on }[a,b]\ \Rightarrow\ \exists\,m,M\in[a,b]:\ f(m)\le f(x)\le f(M)\]

A continuous function on a closed and bounded interval \([a,b]\) attains an absolute maximum and minimum. It does not just "approach" them: it actually takes them.

Both hypotheses matter. On \((0,1)\) the function \(1/x\) is unbounded; on \([0,\infty)\) the function \(x\) has no maximum. If closed or bounded is missing, the theorem can fail.

This result justifies looking for extrema in optimization problems: we know they exist before searching.

Think about it

Why does \(f(x)=x\) on \((0,1)\) not contradict the theorem?

Because the interval is open: it attains no maximum or minimum, and the theorem requires a closed interval.

Does \(f(x)=x/(1+|x|)\) attain a maximum on \(\mathbb R\)?

No. It is continuous and bounded by \(1\), but \(f(x)<1\) for all \(x\) and \(f(x)\to1\) as \(x\to\infty\). The domain is not bounded.

Exercises 20

  1. For which of these functions is an absolute maximum NOT guaranteed by the extreme value theorem?
  2. Which condition of the extreme value theorem fails for \(f(x)=\dfrac{1}{x}\) on \([-1,1]\)?
  3. What does the extreme value theorem guarantee for a function continuous on \([a,b]\)?
  4. Which of these intervals is closed and bounded?
  5. What does it mean for \(f\) to have an absolute maximum on \([a,b]\)?
  6. Which of these functions is NOT continuous on \([0,1]\)?
  7. Let \(f(x)=x^{3}-3x\) on \([-2,2]\). What is the absolute maximum value of \(f\) on that interval?
  8. Which of these functions shows that the theorem fails when the function is not continuous?
  9. What does it mean for an interval to be bounded?
  10. Let \(f(x)=x^{2}\) be continuous on \([0,3]\). What is the absolute maximum of \(f\) on \([0,3]\)?
  11. Find the maximum and minimum of \(f(x)=x^2\) on \([-1,2]\).
  12. What is the maximum of \(f(x)=x^2-4x\) on \([0,5]\)?
  13. Why does the theorem not apply to \(f(x)=1/x\) on \((0,1]\), even though \(f\) is continuous there?
  14. Let \(g(x)=x\) on \([0,1)\). Which statement is true?
  15. Let \(f(x)=\cos x\) on \([0,\pi]\). At which points does it attain its maximum and minimum?
  16. If \(f\) is continuous on \([a,b]\), what immediate consequence does the theorem have?
  17. In the proof one takes \((x_n)\) in \([a,b]\) with \(f(x_n)\to\sup f\). Which theorem gives a convergent subsequence of \((x_n)\)?
  18. Find the maximum and minimum of \(f(x)=x^3-3x\) on \([-2,3]\).
  19. Let \(f(x)=x\) if \(0\le x<1\) and \(f(1)=0\), defined on \([0,1]\). Which hypothesis of the theorem fails and what happens?
  20. Let \(f\) be continuous on \([a,b]\) and \(M=\sup\{f(x):x\in[a,b]\}\). What is true?

What comes next

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.
  • Michael Spivak, Calculus, 4.ª edición, Publish or Perish, 2008.
  • Stephen Abbott, Understanding Analysis, 2nd edition, Springer, 2015. §4.4, Teorema 4.4.3
  • Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976. Cap. 4, Teorema 4.16
  • Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th edition, Wiley, 2011. §5.3, Teorema 5.3.4