Exercise · ZW-ROLLES-THEOREM-020 · ★★★ Advanced
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 \(f'\)?
Choose one option.
A That \(f'\) is positive before \(c\) and negative after. B Nothing: it only ensures an interior point where \(f'=0\). C That \(f'\) changes sign at \(c\). D That \(f'\) is positive on all of \((a,b)\). Show answer B. Nothing: it only ensures an interior point where \(f'=0\).← Back to the question
Explanation The theorem only ensures the existence of a zero of \(f'\). It says nothing about the sign: \(f'\) can touch zero without changing sign, as for \(x^3\) at \(c=0\) (although \(x^3\) does not satisfy \(f(a)=f(b)\) on \([-1,1]\)).
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-008 Which of these cases is continuous with f(a)=f(b), but has no c∈(a,b) with f'(c)=0?