Question

$Letf:[-\frac{1}{3},3]\to Randg:[-\frac{1}{3},3]\to Rdefinedbyf\left(x\right)=[x^2-4]andg\left(x\right)=|x-2|f\left(x\right)+|3x-5|f\left(x\right)$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x$ for $x\in R,$ then

• A. f is discontinuous exactly at eight points in $[-\frac{1}{3},3]$
• B. f is discontinuous exactly at nine points in $[-\frac{1}{3},3]$
• C. g is discontinuous exactly at ten points in $[-\frac{1}{3},3]$
• D. g is discontinuous exactly at nine points in $[-\frac{1}{3},3]$

Answer 1

Answer: (B), (C)

The correct options are
B f is discontinuous exactly at nine points in $[-\frac{1}{3},3]$
C g is discontinuous exactly at ten points in $[-\frac{1}{3},3]$

$g\left(x\right)=[x^2-4](|x-2|+|3x-5|)$
For $x\in [-\frac{1}{3},3],(x^2-4)\in [-4,5]$
So $f\left(x\right)=[x^2-4]$ is discontinuous at 9 points
$g\left(x\right)$ is not differentiable at 9 points of $f\left(x\right)$ and $x=\frac{5}{3}.$
So $g\left(x\right)$ is not differentiable at 10 points.