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
$f(x) = \left\{ \begin{array}{l}\frac{{{x^3} - 2{x^2} - 5x + 6}}{{x - 1}}\,\,\,if\,\,x\, \ne 1,\,\,on\,[ - 2,3]\\ - 6\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,x\, = 1\end{array} \right.$
Similar Questions
Consider a quadratic equation $ax^2 + bx + c = 0,$ where $2a + 3b + 6c = 0$ and let $g(x) = a\frac{{{x^3}}}{3} + b\frac{{{x^2}}}{2} + cx.$
Statement $1:$ The quadratic equation has at least one root in the interval $(0, 1).$
Statement $2:$ The Rolle's theorem is applicable to function $g(x)$ on the interval $[0, 1 ].$
Consider a quadratic equation $ax^2 + bx + c = 0,$ where $2a + 3b + 6c = 0$ and let $g(x) = a\frac{{{x^3}}}{3} + b\frac{{{x^2}}}{2} + cx.$
Statement $1:$ The quadratic equation has at least one root in the interval $(0, 1).$
Statement $2:$ The Rolle's theorem is applicable to function $g(x)$ on the interval $[0, 1 ].$
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