Applications of derivatives · Definition

Rolle's theorem

calculusapplicationstheorems

Before you start you need to know

Rolle's theorem guarantees a horizontal tangent inside the interval when \(f\) takes the same value at both endpoints.

Definition

If \(f\) is continuous on \([a,b]\), differentiable on \((a,b)\) and \(f(a)=f(b)\), there is at least one \(c\in(a,b)\) with \(f^{\prime}(c)=0\).

\[f(a)=f(b)\ \Rightarrow\ \exists\,c\in(a,b):\ f'(c)=0\]

Continuity on \([a,b]\) guarantees that \(f\) attains a maximum and a minimum. If one of them lies in \((a,b)\), Fermat's theorem forces \(f'=0\) there; if both lie at the endpoints, \(f\) is constant and \(f'=0\) everywhere.

Guide example: for \(f(x)=x^{2}-4x+3\) on \([1,3]\), \(f(1)=f(3)=0\) and \(f'(x)=2x-4\). Solving \(2c-4=0\) gives \(c=2\), which lies in \((1,3)\).

Typical error: forgetting that \(f\) must be differentiable on \((a,b)\). For \(f(x)=|x|\) on \([-1,1]\) the function is continuous with equal endpoint values, but it is not differentiable at \(0\) and no \(c\) gives \(f'(c)=0\).

Think about it

Which three hypotheses does Rolle’s theorem need?

Continuity on [a,b], differentiability on (a,b) and f(a)=f(b).

Can Rolle be applied to \(f(x)=\tan x\) on \([0,\pi]\) even though \(f(0)=f(\pi)=0\)?

No. \(\tan x\) is undefined at \(\pi/2\), so it is not continuous on \([0,\pi]\). In fact \(f'(x)=\sec^2x\ge1\) never vanishes.

Exercises 20

  1. Verify Rolle's theorem for \(f(x)=x^{2} - 4 x + 3\) on \([1,3]\) and find c with \(f'(c)=0\).
  2. Verify Rolle's theorem for \(f(x)=\sin x\) on \([0,\pi]\) and find c with \(f'(c)=0\).
  3. Which of these conditions is NOT a hypothesis of Rolle's theorem?
  4. In which of these cases do the hypotheses of Rolle's theorem NOT hold?
  5. If the hypotheses of Rolle's theorem hold on \([a,b]\), what does the conclusion state?
  6. For \(f(x)=x^{3}-x\) on \([-1,1]\), which of these values satisfies \(f'(c)=0\) with \(c\in(-1,1)\)?
  7. For \(f(x)=x^{2}\) on \([0,2]\), which \(c\in(0,2)\) satisfies \(f'(c)=\frac{f(2)-f(0)}{2-0}\)?
  8. Which of these cases is continuous with \(f(a)=f(b)\), but has no \(c\in(a,b)\) with \(f'(c)=0\)?
  9. What does Rolle's theorem mean geometrically?
  10. For \(f(x)=x^{2}-6x+5\) on \([1,5]\), which value of \(c\in(1,5)\) satisfies \(f'(c)=0\)?
  11. For \(f(x)=x^2-4x+3\) on \([1,3]\), which point \(c\) does Rolle's theorem guarantee?
  12. For \(f(x)=\sin x\) on \([0,\pi]\), at which point does \(f'\) vanish?
  13. Which hypothesis of Rolle's theorem fails for \(f(x)=x\) on \([0,1]\)?
  14. For \(f(x)=x^3-x\) on \([0,1]\), what is the point \(c\in(0,1)\) with \(f'(c)=0\)?
  15. For \(f(x)=x^3-3x\) on \([-\sqrt3,\sqrt3]\), which values of \(c\) satisfy \(f'(c)=0\)?
  16. For \(f(x)=\sqrt{1-x^2}\) on \([-1,1]\), does Rolle's theorem apply and which point does it give?
  17. Let \(f(x)=x\) if \(0\le x<1\) and \(f(1)=0\). Why does Rolle's conclusion fail on \([0,1]\)?
  18. A function differentiable on \(\mathbb R\) has three distinct zeros. At least how many distinct zeros does \(f'\) have?
  19. Let \(p(x)=x^3+3x+1\). How many real roots does \(p\) have?
  20. If \(f\) is continuous on \([a,b]\), differentiable on \((a,b)\) and \(f(a)=f(b)\), what else does the theorem say about the sign of \(f'\)?

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. §5.3, Teorema 5.3.1
  • Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976. Cap. 5, Teorema 5.10
  • Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th edition, Wiley, 2011. §6.2, Teorema 6.2.3