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

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
If $f(x) = \left\{ \begin{array}{l}a{x^2} + b;\,\,x \le 0\\\,\,\,\,\,\,\,\,\,{x^2};x > 0\,\end{array} \right.$ possesses derivative at $x = 0$, then
  • A
    $a = 0,b = 0$
  • B
    $a > 0, = 0$
  • ✓
    $a \in R, = 0$
  • D
    None of these
✓

Answer

Correct option: C.
$a \in R, = 0$
c
(c) $f(x)$ possesses derivative at $x = 0$, so it is both continuous and differentiable at $x = 0$.

Now $f(0 + 0) = 0$, $f(0 - 0) = b,f(0) = b$, 

$\therefore \,\,b = 0$

Also $Rf'(0) = 0,Lf'(0) = 0,\forall a \in R$

$\therefore \,f'(0) = 0$ if $b = 0$.

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