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(1) $f:\mathbb R\to \mathbb R$ is $C^\infty$.

(2) $f^2:\mathbb R\to \mathbb R$ is $C^\infty$.
EDIT: The right formulation is: If $f\ge 0$ is $C^\infty$, then one can choose a square root of $f$ which is twice differentiable, but not better in general.

(3) $f^2$ and $f^3:\mathbb R\to \mathbb R$ are both $C^\infty$.

(4) $f^p$ and $f^q:\mathbb R\to \mathbb R$ are both $C^\infty$. EDIT: Where $p$, $q$ are relatively prime.

Obviously, (1) $\implies$ (2), (3), (4).

(2) $\implies$ $f$ is twice differentiable, in general not better. See:
Dmitri Alekseevky, Andreas Kriegl, Mark Losik, Peter W. Michor: Choosing roots of polynomials smoothly, Israel J. Math 105 (1998), p. 203-233.(pdf)

(3) $\implies$ (1). See:
MR0682456 Joris, Henri: Une $C^\infty$-application non-immersive qui possède la propriété universelle des immersions. Arch. Math. (Basel) 39 (1982), no. 3, 269–277.
and:
MR2179865 Myers, Robert: An elementary proof of Joris's theorem. Amer. Math. Monthly 112 (2005), no. 9, 829–831.

(4) $\implies$ (1). See:
MR0833407 Duncan, John; Krantz, Steven G.; Parks, Harold R.: Nonlinear conditions for differentiability of functions. J. Analyse Math. 45 (1985), 46–68.

(1) $f:\mathbb R\to \mathbb R$ is $C^\infty$.

(2) $f^2:\mathbb R\to \mathbb R$ is $C^\infty$.

(3) $f^2$ and $f^3:\mathbb R\to \mathbb R$ are both $C^\infty$.

(4) $f^p$ and $f^q:\mathbb R\to \mathbb R$ are both $C^\infty$.

Obviously, (1) $\implies$ (2), (3), (4).

(2) $\implies$ $f$ is twice differentiable, in general not better.
See:
Dmitri Alekseevky, Andreas Kriegl, Mark Losik, Peter W. Michor: Choosing roots of polynomials smoothly, Israel J. Math 105 (1998), p. 203-233.(pdf)

(3) $\implies$ (1). See:
MR0682456 Joris, Henri: Une $C^\infty$-application non-immersive qui possède la propriété universelle des immersions. Arch. Math. (Basel) 39 (1982), no. 3, 269–277.
and:
MR2179865 Myers, Robert: An elementary proof of Joris's theorem. Amer. Math. Monthly 112 (2005), no. 9, 829–831.

(4) $\implies$ (1). See:
MR0833407 Duncan, John; Krantz, Steven G.; Parks, Harold R.: Nonlinear conditions for differentiability of functions. J. Analyse Math. 45 (1985), 46–68.