Question

Let $f :\left[-\frac{1}{2}, 2\right] \rightarrow R$ and $g :\left[-\frac{1}{2}, 2\right] \rightarrow R$ be functions defined by $f ( x )=\left[ x ^{2}-3\right]$ and $g ( x )=| x | f ( x )+14 x -7 \mid f ( x )$ where [y] denotes the greatest integer less than or equal to $y$ for $y \in R$. Then

• A. $f$ is discontinuous exactly at three points in $\left[-\frac{1}{2}, 2\right]$
• B. $f$ is discontinuous exactly at four points in $\left[-\frac{1}{2}, 2\right]$
• C. $g$ is NOT differentiable exactly at four points in $\left(-\frac{1}{2}, 2\right)$
• D. $g$ is NOT differentiable exactly at five points in $\left(-\frac{1}{2}, 2\right)$

Answer 1

Answer: (B), (C)

$ f :\left[-\frac{1}{2}, 2\right] \rightarrow R $
$ f ( x )=\left[ x ^{2}-3\right]$
is discontinuous at four points $x \in\left[-\frac{1}{2}, 2\right]$
$x =1, \sqrt{2}, \sqrt{3}\, \& \,2$
Hence not differentiable so $( B )$ is correct
also now, $g(x)=\underset{\overset{\downarrow}{0}}{|x|} \cdot f(x)+|\underset{\overset{\downarrow}{7/4}}{4 x}-7| f(x) $
when $x \in\left(-\frac{1}{2}, 2\right)$ at $x=\frac{7}{4} g(x)$ is continuous and $g(x)$ is discontinuous at $4$ points $0,1, \sqrt{2}, \sqrt{3}$.
Hence not differentiable at $4$ points so option (C) is correct.