Question

Number of point(s) in $[1,2]$ where $f(x)=(-2)^{\left[x^{3}\right]}$ is non-differentiable (where $[\,\,.\,\,]$ denotes greatest integer function)

Answer 1

$1 \leq x^{3} \leq 8$
discontinuous at $x^{3}=2,3,4,5,6,7$; hence non differentiable
$f(1)=-2$
$f(1+h)=-2$ continuous and differentiable at
$x=1$
$f(8)=(-2)^{8^{3}} \& f(2-h)$
$=\displaystyle\lim _{h \rightarrow 0}(-2)^{[(8-h)]}=(-2)^{7}$
Non-differentiable for $x^{3}=8$
Total points of non-differentiablity are $7$ .