We analyze the class of increasing utility functions exhibiting all derivatives of alternating sign. This property, that we call mixed risk aversion, is satisfied by the utility functions most commonly used in financial economics. The utility functions displaying mixed risk aversion can be expressed as mixtures of exponential functions. We characterize stochastic dominance and we find conditions for both mutual aggravation and mutual amelioration of risks when agents are mixed risk averse. Finally, the analysis of the distribution function describing a mixed utility allows one to characterize the behaviour of its indexes of risk aversion and to discuss its implications for portfolio selection.