Let [x] stand for greatest integer function and f ( x ) = 4 x^2 ,… - Vidyadip

MCQ

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 $b$.

✓

Answer

Correct option: C.

$ƒ(x)$ is continuous and differentiable in $\left( {\frac{{ - 1}}{2},\frac{1}{2}} \right)\forall \,a\,\, \in R\,\& \,b\, = 1$

c
$f(x) = \left\{ {\begin{array}{*{20}{c}}
{4{x^2} - x,}&{ - \frac{1}{2} \le x < 0}\\
{a{x^2} - bx,}&{0 \le x < \frac{1}{2}}
\end{array}} \right.$

$f^{\prime}\left(0^{-}\right)=-1, f^{\prime}\left(0^{+}\right)=-b$

$\Rightarrow$ if $\mathrm{b}=1 \Rightarrow f(\mathrm{x})$ is differentiable.

and $f(\mathrm{x})$ is continuous always.

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