I know of a proof for this which involves finding matrix of second derivatives (Hessian) for the given expression and proving that it is negative semi definite. But that is quite sophisticated for my use. I need to prove it using the fact that the sum of concave functions is a concave function (or another easier method). How do I go about it?
closed as off-topic by Henrik, Jendrik Stelzner, Delta-u, José Carlos Santos, Gibbs Sep 6 '18 at 19:15
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Henrik, Jendrik Stelzner, Delta-u, José Carlos Santos, Gibbs
Everything you have in sight there are linear (affine) functions, their sums and composition with the function $s(x)=\log(1+e^x)$. So all you need to show is that $s$ is convex, which is a simple exercise in one variable.