Question

Let $f:[-\frac{1}{2},2]\to R$ and $g:[-\frac{1}{2},2]\to R$ be functions defined by
$f\left(x\right)=[x^2-3]\text{and}g\left(x\right)=|x|f\left(x\right)+|4x-7|f\left(x\right)$,where [y] denotes the greatest integer less than or equal to y for $yϵR$. Then

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

Answer 1

Answer: (B), (C)

The correct options are
B f is discontinuous exactly at four points in $[-\frac{1}{2},2]$
C g is not differentiable exactly at four points in $(-\frac{1}{2},2)$

f and g: $[-\frac{1}{2},2]\to R$
$f\left(x\right)=[x^2-3]\text{and}g\left(x\right)=|x|f\left(x\right)+|4x-7|f\left(x\right)$
$[x^2-3]$ is discontinuous at all integral points in
$[-\frac{1}{2},2]$ which happens at $x=1,x=\sqrt{2},x=\sqrt{3},x=2$
F is discontinuous at 4 points.
$g\left(x\right)=(|x|+|4x-7|)f\left(x\right)$
F is not differentiable at $x=1,\sqrt{2},\sqrt{3}$
And $|x|+|4x-7|$ is not differentiable at x=0
$x=\frac{7}{4}$
$f\left(x\right)=0$ in $[\sqrt{3},2]$
So at $\frac{7}{4}ϵ[\sqrt{3},2]$ the f(x) is differentiable
Hence g(x) is not differentiable at 4 points