In this work we provide the best estimate for the functional coefficient in a linear model with functional response and multivariate predictor, exploiting fully the information provided by both functions and derivatives. The model includes possible random effects due to repeated measurements. We also study the theory of optimal design of experiments when functional observations occur. We define different optimality criteria for the estimate of the functional coefficient. We provide a strong theoretical foundation to prove that the computation of these optimal designs, in the case of linear models, is the same as in the classical theory, but different interpretations need to be given.