The number of points, where the function f:RR, [f(x)=|x-1||x-2||x-1|+(x-3)|x^2-5x+4|] is N…

Question

The number of points, where the function $f:\mathbb{R}\to\mathbb{R}$,
$$f(x)=|x-1|\cos|x-2|\sin|x-1|+(x-3)|x^2-5x+4|$$
is NOT differentiable, is:

Accepted Answer

Note $x^2-5x+4=(x-1)(x-4)$, so $|x^2-5x+4|=|x-1||x-4|$. Thus
$$f(x)=|x-1|\Big[\cos|x-2|\sin|x-1|+(x-3)|x-4|\Big]$$
This factors out $|x-1|$, which has a sharp corner at $x=1$, and the bracket contains $|x-4|$, which has a corner at $x=4$. Checking that the bracket does not vanish at $x=1$ and that the outer factor does not cancel the corner at $x=4$, $f$ is not differentiable at $x=1$ and $x=4$ --- exactly 2 points.

Solutions · 1