Linear and logistic regression output with same neural network

it seems like this should be a very common task, but I have not found anything useful in my research: How can I do linear and logistic regression with the same neural network?

By example, what I mean is the following: Let’s say I want to train a network on pictures of faces and I want a two-dimensional output: The gender and the age of the person on the photo. So for the gender, I would want to do logistic regression, while linear regression for the age.

Should there be two different activation functions in the output layer (sigmoid and linear)? How to deal with the fact that, for convexity reasons, one usually uses MSE loss for linear and logistic loss for logistic regression?

Thanks!