Does $f(0) = 0 \implies f'(x) = 0$?
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Let $f(x)=x$ then $f(0)=0$ but $f'(x)=1$ so not true for all cases.
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Anmol Bhullar
Wannabe mathematician. Currently: Undergraduate in Math/CS.
Updated on August 01, 2022Comments

Anmol Bhullar 3 months
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?

Theodoros Mpalis over 6 yearsindeed. Because 0 is differentiable.

Ian Miller over 6 yearsDoes $f(x)=0 \forall x$ or are you meaning at a specific point $x$?

Anmol Bhullar over 6 years@IanMiller sorry I made a typo I meant to say $f(0) = 0$ rather than $f(x) = 0$

Ian Miller over 6 yearsWhats the question in the book where it says $f(0)=f'(0)=0$. Some context will help us answer your problem.

Thomas Andrews over 6 yearsOkay, with the new edit, no, it is possible for $f(0)=0$ but $f'(0)\neq 0$.


Archis Welankar over 6 yearsAccording to your edit yes as 0 is constant

Theodoros Mpalis over 6 yearsI think he mean that $f$ is constant.

Archis Welankar over 6 yearsYes so i wrote my comment