Exercise · ZW-ROLLES-THEOREM-016 · ★★☆ Intermediate
For \(f(x)=\sqrt{1-x^2}\) on \([-1,1]\), does Rolle's theorem apply and which point does it give?
Choose one option.
A No, because \(f\) is not differentiable at \(\pm1\). B Yes, with \(c=0\), even though \(f\) is not differentiable at \(\pm1\). C Yes, with \(c=\pm1\). D No, because \(f(-1)\ne f(1)\). Show answer B. Yes, with \(c=0\), even though \(f\) is not differentiable at \(\pm1\).← Back to the question
Explanation It is continuous on \([-1,1]\), differentiable on \((-1,1)\) and \(f(\pm1)=0\). Differentiability is only required inside; \(f'(x)=-x/\sqrt{1-x^2}\) vanishes at \(0\).
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?