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For my master thesis, I have to show/prove that the log-likelihood function for logistic regression is globally concave. My supervisor told me that one way to show this is to use the fact that $X'X$ is positive definite and using this I can show that the Hessian of the log-likelihood is negative definite. I managed to show that if $X$ is of full rank then $X'X$ is positive definite. However, I don't know how to proceed from here and would very much appreciate some help.

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