Question

For which of the following functions is Rolle's theorem not applicable?

• A. \(f(x)=\log (x^{2}+2)-\log 3,on[-1,1]\)
• B. \(f(x)=3+(x-1)^{2/3},on[0,3]\)
• C. \(f(x)=e^{1-x^{2}},on[-1,1]\)
• D. \(f(x)=\sqrt{4-x^{2}},on[-2,2]\)

Hint:

Check continuity, differentiability and equal end values for each function.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Rolle's theorem needs \(f\) continuous on \([a,b]\), differentiable on \((a,b)\) and \(f(a)=f(b)\).

Step 2: Check the functions
\(f(x)=\log (x^{2}+2)-\log 3\) and \(e^{1-x^{2}}\) are even, smooth, with equal end values on \([-1,1]\). \(\sqrt{4-x^{2}}\) is continuous on \([-2,2]\), differentiable on \((-2,2)\) and \(f(\pm 2)=0\).

Step 3: The failing function
For \(f(x)=3+(x-1)^{2/3}\), \(f'(x)=\dfrac{2}{3}(x-1)^{-1/3}\) does not exist at \(x=1\), which lies in \((0,3)\).

Step 4: Conclusion
Rolle's theorem is not applicable to \(3+(x-1)^{2/3}\) on \([0,3]\). Also \(f(0)=4\neq f(3)=3+2^{2/3}\).

Final Answer:
Rolle's theorem fails for \(f(x)=3+(x-1)^{2/3}\) on \([0,3]\). \[ \boxed{\text{(B) }f(x)=3+(x-1)^{2/3},\text{ on }[0,3]} \]