Can't move it, can't hold it, and can't slow it down

Muscles have to be exposed to more than they are used to handling if there is to be a positive change. Hopefully that is done in a safe manner. Confronted with a state of fatigue that is beyond what the body is used to, the body, as self-protection, will make a positive adaption by becoming stronger if given enough recovery time.

There are three stages of fatigue associated with resistance training. When you can no longer lift or move a weight you've reached concentric or positive failure. When you can no longer hold the weight you've reached static failure. This produces a deeper fatigue than positive failure. When you can no longer stop a weight from falling or lowering you've reached negative or eccentric failure.  This is the deepest fatigue.  Eccentric failure is best conducted with a trainer or spotter and on equipment where it can be safely performed.

Negative reps should be used sparingly. Negatives put a much bigger hit on the system. You're basically pulling the muscles out of a contraction. Micro-trauma to the muscles occurs. If not done safely you can have serious injury. One way to do this is to complete a regular set and then have the trainer immediately lift the weight for the client and have the client lower the weight slowly.

If one performs chin-ups to the point where she just barely gets the last one up that's positive failure. If she continues to hold at the top position until she can no longer hold it that is static failure. After that if she fights the lowering of her body weight with every fiber of her being she will achieve negative failure.

Achieving negative failure? That is far from a negative thing. You have successfully stressed the body to the point where the body as an act self-protection must make a positive adaption to survive.

Conducted correctly negatives can be very safe. I am currently conducting physical therapist prescribed negative reps with a client with two knee replacements through a limited range. There are a lot of factors to consider.  The type of exercise, the range of motion, pre-existing conditions are just a few of those considerations.  At Austin Personal Trainers and New Orleans Fitness Training we have MedX rehabilitative exercise equipment that is specially suited for concentric, static and eccentric reps.