# Is there an infinitely differentiable function such that $f$ intersects the $x$ axis only at the origin and no derivative of $f$ is $0$?

Is there an infinitely differentiable function $f$ such that

• $f(x)=0 \iff x=0$,
• $f$ is infinitely differentiable
• No derivative of $f$ is ever equal to zero.
• Think about what you can do with the exponential function. Oct 10, 2014 at 18:23

Consider the function $f:\Bbb R \to \Bbb R$ s.t $f(x)=e^x-1$.