In which of the following functions is Rolle's theorem applicable ?
In which of the following functions is Rolle's theorem applicable ?
- A
$f(x) = \left\{ \begin{array}{l}
x,\,\,\,\,\,\,\,0 \le x < 1\\
0,\,\,\,\,\,\,\,x = 0\,\,\,\,\,\,
\end{array} \right.on\,\,\left[ {0,1} \right]$ - B
$f(x) = \left\{ \begin{array}{l}
\frac{{\sin x}}{x},\,\,\,\,\,\,\, - \pi \le x < 0\\
0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x = 0\,\,\,\,\,\,
\end{array} \right.on\,\,\left[ { - \pi ,0} \right]$ - C
$f(x) = \frac{{{x^2} - x - 6}}{{x - 1}}\,\,\,\,\,on\left[ { - 2,3} \right]$
- D
$f(x) = \left\{ \begin{array}{l}
\frac{{{x^3} - 2{x^2} + 5x + 6}}{{x - 1}},\,\,\,\,if\,\,x \ne 0\,\,\,\,\,\,\\
- 6,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,x = 1\,\,\,\,\,\,\,\,\,\,\,\,\
\end{array} \right.on\left[ {-2,3} \right]$
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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 $f(4)=\frac{1}{4}$.
Consider the following two statements:
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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 $f(4)=\frac{1}{4}$.
Consider the following two statements:
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Given $f (x) =4\,\, - \,\,{\left( {\frac{1}{2}\, - \,x} \right)^{2/3}}\,$ $g (x) = \left\{ \begin{array}{l}\frac{{\tan \,\,[x]}}{x}\,\,\,\,,\,\,x \ne \,0\\1\,\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,x\, = \,0\end{array} \right.$
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