In which of the following functions Rolle’s theorem is applicable ?
In which of the following functions Rolle’s theorem is applicable ?
- A
$ f(x) =\left\{ \begin{array}{l}x\,\,\,,\,\,0\, \le \,x\, < \,\,1\\0\,\,\,\,,\,\,\,\,\,\,\,\,\,x\,\, = 1\end{array} \right.$ on $[0, 1]$
- B
$f(x) = \left\{ \begin{array}{l}\frac{{\sin x}}{x}\,\,,\, - \pi \, \le x\, < 0\\\,0\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\,\,\,x\, = \,0\end{array} \right.$ on $[-\pi , 0]$
- C
$f(x)= \frac{{{x^2} - x - 6}}{{x - 1}}$ on $[-2,3]$
- D
$f(x) = \left\{ \begin{array}{l}\frac{{{x^3} - 2{x^2} - 5x + 6}}{{x - 1}}\,\,\,if\,\,x\, \ne 1,\,\,on\,[ - 2,3]\\ - 6\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,x\, = 1\end{array} \right.$
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Let $f :[2,4] \rightarrow R$ be a differentiable function such that $\left(x \log _e x\right) f^{\prime}(x)+\left(\log _e x\right) f(x)+f(x) \geq 1$, $x \in[2,4]$ with $f(2)=\frac{1}{2}$ and $f(4)=\frac{1}{4}$.
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