Derivative Function Exercise

Question

If $f(x)=g(x)\cdot(x-1)^2$ then I need to find a $g(x)$ such that there will not be a second derivative of $f$ for $x=1$. I don't know what to search for.

Answer 1

HINT: Consider $g(x)=\operatorname{sgn} (x-1)$

Answer 2

Can you come up with such a function $f$?

Note that $f(1)$ has to vanish, if the equation between $f$ and $g$ has to hold for all $x$.

Then $g(x)=f(x)/(x-1)^2$ for $x \ne 1$.

The last thing to do is probably a good choice for $g(1)$.