MCQ

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Let $[x]$ stand for greatest integer function and $f\left( x \right) = \left\{ \begin{gathered}
  4{x^2}\, + \,\left[ {2x} \right]x,\,\,if\,x \in \left[ {\frac{{ - 1}}{2}},0 \right) \hfill \\
  a{x^2}\, - \,bx,\,\,\,\,\,\,\,\,\,if\,x \in \left[ {0,\frac{1}{2}} \right) \hfill \\ 
\end{gathered}  \right.$ then

• A
  $ƒ(x)$ is continuous  in $\left( {\frac{{ - 1}}{2},\frac{1}{2}} \right)$ , iff $a = 4$ and $b = 0$.
• B
  $ƒ(x)$ is continuous and differentiable in $\left( {\frac{{ - 1}}{2},\frac{1}{2}} \right)$ iff $a = 4$, $b = 1$.
• ✓
  $ƒ(x)$ is continuous and differentiable in $\left( {\frac{{ - 1}}{2},\frac{1}{2}} \right)\forall \,a\,\, \in R\,\& \,b\, = 1$
• D
  $ƒ(x)$ is not differentiable in $\left( {\frac{{ - 1}}{2},\frac{1}{2}} \right)$ for any value of $a$ and $b$.