Exercise · ZW-ROLLES-THEOREM-008 · ★★★ Advanced
Which of these cases is continuous with \(f(a)=f(b)\), but has no \(c\in(a,b)\) with \(f'(c)=0\)?
Choose one option.
Show answer
- C. \(f(x)=|x|\) on \([-1,1]\)
Explanation
The function \(|x|\) is continuous with \(f(-1)=f(1)=1\), but it is not differentiable at \(0\), and elsewhere its derivative equals \(\pm 1\), never \(0\). The other cases either have a valid \(c\) or fail \(f(a)=f(b)\).
Review the concept: Rolle's theorem
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 \(f^{\prime}(c)=0\).
\[f(a)=f(b)\ \Rightarrow\ \exists\,c\in(a,b):\ f'(c)=0\]
More practice problems: rolle's theorem
- ZW-ROLLES-THEOREM-001 Verify Rolle's theorem for f(x)=x^2 - 4 x + 3 on [1,3] and find c with f'(c)=0.
- ZW-ROLLES-THEOREM-002 Verify Rolle's theorem for f(x)=sin x on [0,π] and find c with f'(c)=0.
- ZW-ROLLES-THEOREM-003 Which of these conditions is NOT a hypothesis of Rolle's theorem?
- ZW-ROLLES-THEOREM-004 In which of these cases do the hypotheses of Rolle's theorem NOT hold?
- ZW-ROLLES-THEOREM-005 If the hypotheses of Rolle's theorem hold on [a,b], what does the conclusion state?
- ZW-ROLLES-THEOREM-006 For f(x)=x^3-x on [-1,1], which of these values satisfies f'(c)=0 with c∈(-1,1)?
- ZW-ROLLES-THEOREM-007 For f(x)=x^2 on [0,2], which c∈(0,2) satisfies f'(c)=(f(2)-f(0))/(2-0)?
- ZW-ROLLES-THEOREM-009 What does Rolle's theorem mean geometrically?