Can a function be approximated by an infinitly often differentiable function with common values at the boundary of the Intervall

Question

Let $ f:[0,1] \to \mathbb{R} $ be a $ C^1 $-function. Does for every $ \epsilon > 0 $ exist an $C^{\infty}-$function $ g:[0,1]\to \mathbb{R} $ such that $ g(0)= f(0) $, $ g(1) = f(1) $ and $ \vert \vert g-f \vert \vert_{C^1} < \epsilon $, where $ \vert \vert f \vert \vert_{C^1}:= \vert \vert f \vert \vert_{C^0} + \vert \vert f' \vert \vert_{C^0} $. If yes, is there any reference?