I'm looking at the derivative of |-5x+8| WRT x
The critical point is 8/5. The derivative is = -5 if x<8/5 and 5 if x>8/5
this shows that the lim f'(x) as x-> 8/5 from above 8/5 is 5
and lim f'(x) as x-> 8/5 from below 8/5 is -5
This, in turn, shows that the derivative is discontinuous at 8/5 and therefore the function is not derivable at that point.
Does the above constitute a proof? Have I shows that the function is not derivable at its critical point? Have I successfully proven the limit does not exist?