Question

Let f(x)=\left\\{\begin{matrix} |4x^2-8x+5|, & if\,8x^2-6x+1\geq0 \\\ |4x^2-8x+5|, & if\,8x^2-6x+1<0 \end{matrix}\right. where [α] denotes the greatest integer less than or equal to α.Then the number of points in R where f is not differentiable is

Answer 1

f(x)=\left\\{\begin{matrix} |4x^2-8x+5|, & if\,8x^2-6x+1\geq0 \\\ |4x^2-8x+5|, & if\,8x^2-6x+1<0 \end{matrix}\right.

=\left\\{\begin{matrix} |4x^2-8x+5|, & if\,x\in [-\infty,\frac{1}{4}]\bigcup[\frac{1}{2},\infty] \\\ |4x^2-8x+5|, & if\, x\in(\frac{1}{4},\frac{1}{2})\end{matrix}\right.

f(x)=f(x)=\left\\{\begin{matrix} 4x^2-8x+5 &if\,x\in [-\infty,\frac{1}{4}]\bigcup[\frac{1}{2},\infty] & \\\ 3&x\in(\frac{1}{4},\frac{2-\sqrt2}{2}) & \\\ 2& x\in(\frac{2-\sqrt2}{2},\frac{1}{2}) & \end{matrix}\right.

∴ Non-diff at

x=$\frac{1}{4}$,$\frac{2-\sqrt2}{2}$,$\frac{1}{2}$