Is extra mixing really needed in asymptotic giant branch stars?

We demonstrate that the amount of extra mixing required to fit the observed low C/N and ¹²C/¹³C ratios in first giant branch (FGB) stars is also sufficient to explain the carbon and nitrogen abundances of Galactic asymptotic/second giant branch (AGB) stars. We simulate the effect of extra mixing on the FGB by setting the composition of the envelope to that observed in low-mass (M ≤ 2 M) FGB stars, and then evolve the models to the tip of the AGB. The inclusion of FGB extra mixing compositional changes has a strong effect on the C and N abundance in our AGB models, leading to compositions consistent with those measured in Galactic carbon-rich stars. The composition of the models is also consistent with C and N abundances measured in mainstream silicon carbide grains. While our models cover the range of C abundances measured in carbon stars in the LMC cluster NGC1846, we cannot simultaneously match the composition of the O- and C-rich stars at the same time. A second important result is that our models only match the oxygen isotopic composition of K and some M, MS giants, and are not able to match the oxygen composition of carbon-rich AGB stars. By increasing the abundance of ¹⁶O in the intershell (based on observational evidence) it is possible to reproduce the observed trend of increasing ¹⁶O/ ¹⁸O and ¹⁶O/¹⁷O ratios with evolutionary phase. We also find that some Li production takes place during the AGB and that Li-rich carbon stars (log ε(Li) ≳ 1) can be produced. These models show a correlation between increasing Li abundances and C. The models cannot explain the composition of the most Li-enriched carbon stars, nor can we produce a Li-rich carbon star if we assume extra mixing occurs during the FGB owing to ³He destruction. We tentatively conclude that (1) if extra mixing occurs during the AGB it likely only occurs efficiently in low metallicity objects, or when the stars are heavily obscured making spectroscopic observations difficult and (2) the intershell compositions of AGB stars need further investigation.