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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
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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