Question

Which of the following functions satisfies all conditions of the Rolle’s theorem in the intervals specified?

• A. $f\left(x\right)=x^{\frac{1}{3}},x\in \left[- 2,3\right]$
• B. $f\left(x\right)=sin x,x\in \left[- \pi , \frac{\pi }{6}\right]$
• C. $f\left(x\right)=ln \left(\frac{x^{2} + a b}{x \left(a + b\right)}\right),$ $x\in \left[a , b\right],0 < a < b$
• D. $f\left(x\right)=e^{x^{2} - x},x\in \left[0,4\right]$

Answer 1

Answer: (C)

$f\left(x\right)=x^{\frac{1}{3}}$ is not differentiable at $x=0$
For $f\left(x\right)=sin x,f\left(- \pi \right)\neq f\left(\frac{\pi }{6}\right)$
For $f\left(x\right)=ln \left(\frac{x^{2} + a b}{x \left(a + b\right)}\right)$
$f\left(a\right)=f\left(b\right)=ln \left(1\right)=0$
Also, it is continuous in $\left[a , b\right]$ and differentiable in $\left(a , b\right)$ . Hence, Rolle’s Theorem is applicable.
For $f\left(x\right)=e^{x^{2} - x},f\left(0\right)\neq f\left(4\right)$