Including database connection file in php

You have to globalize the variable before using in a function.
Just add this line at the top of your function:

function authenticate($Email, $Password)
    {   
         global $db;
         $HashedPassword = password_hash($Password, PASSWORD_DEFAULT);        
         $sql = "SELECT * FROM app_users WHERE user_email='$Email'"; 
         $result = $db->query($sql);
         $User = $result->fetch_object();
    ...

Add global $db; to the beginning of your authenticate function.

However, I strongly recommend you learn about PDO: http://wiki.hashphp.org/PDO_Tutorial_for_MySQL_Developers

Your code is currently unsafe