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
$f(x) = \left\{ \begin{array}{l}
\frac{{{x^3} - 2{x^2} + 5x + 6}}{{x - 1}},\,\,\,\,if\,\,x \ne 0\,\,\,\,\,\,\\
- 6,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,x = 1\,\,\,\,\,\,\,\,\,\,\,\,\
\end{array} \right.on\left[ {-2,3} \right]$
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 $[0, 1 ].$
- [AIEEE 2012]
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