No. she ain't trying her hardest because all she does is fall down and cry each time something goes wrong. It's one thing to break down when it's too much, but i cant take her efforts seriously if she just breaks down after each little thing.
Hell, i stopped taking her seriously when she almost gave up life because Litchi boobs are way bigger than hers.
She never almost gave up on the entire mission over that. If she said that it was clearly a joke.
Yes she gets frustrated over everything but she never completely gives up. If it wasn't for her crying (as I believe that's how she vents), she probably would have given up. This girl isn't full of sadness, but anger and frustration. That is why she goes all emo when she becomes Mu.
I can relate with her, granted I don't cry but I show rage and frustration when every little thing goes wrong. Same emotions, different way of showing it.