Expansion of fermat theorem[in mathematical analysis]

the fermat theorem says:
if a function has it’s extreme value at x, then f'(x)=0
It’s tell us the necessary condition when a funtion has it’s extreme value.But the f'(x) must exist when you want to use this theorem.
 
now I found that:
if a function has it’s extreme value at x, then f’-(x)*f’+(x)<=0
It’s the  necessary condition when a funtion has it’s extreme value.And we do not need f'(x) exist.
 
The proof of above theorem is quite simple, do it by yourselft?
I will publish it after a while.
 
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