Let f(x) = * 20 c x^2 x, ,x > 0 0, , , , , , , , , , , , , ,x = 0… - Vidyadip

MCQ

Let $f(x) = \left\{ {\begin{array}{*{20}{c}}
{{x^2}\ln x,\,x > 0} \\
{0,\,\,\,\,\,\,\,\,\,\,\,\,\,x = 0}
\end{array}} \right\}$, Rolle’s theorem is applicable to $ f $ for $x \in [0,1]$, if $\alpha = $

A
$-2$
B
$-1$
C
$0$
✓
$0.5$

✓

Answer

Correct option: D.

$0.5$

d
(d) For Rolle’s theorem to be applicable to $f,$ for $x \in [0,\,1]$, we should have

$(i)$ $f(1) = f(0)$,

$(ii)$ $f$ is continuous for $x \in [0,\,1]$ and $f$ is differentiable for $x \in (0,\,1)$

From $(i),$ $f(1) = 0$, which is true.

From $(ii),$ $0 = f(0) = f({0_ + }) = \mathop {\lim }\limits_{x \to {0_ + }} {x^\alpha }\ln x$

Which is true only for positive values of $\alpha $, thus $(d)$ is correct.

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