Question

Let $f:\left[-\frac{1}{2},2\right]\to R$ and $g:\left[-\frac{1}{2},2\right]\to R$ be functions defined by $f(x)=\left[x^2-3\right]$ and $g(x)=|x|f(x)+14x-7\mid f(x)$ 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)

$f:\left[-\frac{1}{2},2\right]\to R$
$f(x)=\left[x^2-3\right]$
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, $g(x)=\underset{\stackrel{\downarrow }{0}}{|x|}\cdot f(x)+|\underset{\stackrel{\downarrow }{7/4}}{4x}-7|f(x)$
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.