Does $f(0) = 0 \implies f'(x) = 0$?

Question

Assuming $f(x)$ is differentiable $\forall x$

In my textbook, for one of the questions, it says

$f(0) = f'(0) = 0$, I was a little confused since I

thought $f(0) = 0 \implies f'(x) = 0$ and thought it was

redundant, but I'm probably wrong. If I am, can someone

perhaps give a counterexample?

Answer 1

Let $f(x)=x$ then $f(0)=0$ but $f'(x)=1$ so not true for all cases.

Answer 2

Saying $f(x)=0$ can mean different things depending on context. If the function being defined, then it means that $f(x)=0$ for all $x$, in which case your reasoning is correct. Sometimes, though, it means that for a particular $x$, $f(x)$ takes the value of zero, in which case $f'(x)$ also being zero is notable.

Your recent edit indicates that $f(x)$ equals zero when $x=0$, but not necessarily all $x$. In this case, as in $f(x)=x$, $f(0)=0$, but $f'(x)$ may not be zero.