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: Check the end values
For the second function, $f(0)=3+1=4$ and $f(3)=3+2^{2/3}$, which are not equal.

Step 2: Check differentiability
The derivative has $(x-1)^{-1/3}$, undefined at $x=1$ inside the interval.

Step 3: Compare the options
The other three functions satisfy all conditions, so only this one fails.

Final Answer:
Both the end-value condition and differentiability fail for the second function. \[ \boxed{\text{(B) }f(x)=3+(x-1)^{2/3},\text{ on }[0,3]} \]