<s>[INST] Find the number of points of non differentiability in $(x^2-3x+2)(|x^3-6x^2+11x-6|)$ The expression is factorized as $$(x-2)(x-1)(|(x-1)(x-2)(x-3)|)$$
I expected the points to be $1,2,3$, put looking at the graph, it’s only $x=3$
I kinda figured out the reason for this, because the expression would end up simplifying to
$\pm (x-2)^2(x-1) ^2|x-3|$ but I am not finding this very convincing. What can be a concrete reason for this?
 [/INST] $$F(x)=(x-1)(x-2)|(x-1)(x-2)(x-3)|.$$ Only $x=3$ is the point of non-differentiability. At $x=1,2$ is it differentiable. At these points, we have $(x-a)|x-a|$ which is differentiable at $x=a$.
Note that $g(x)=(x-a)|x-a|= (x-a)(a-x), ~if~ x<a$ and $g(x)=(x-a)^2, ~if ~x\ge a$, hence both the left and right derivatives are equal and zero at $x=a$.
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