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.

Advertisements
(function(g,$){if("undefined"!=typeof g.__ATA){
g.__ATA.initAd({collapseEmpty:'after', sectionId:26942, width:300, height:250});
g.__ATA.initAd({collapseEmpty:'after', sectionId:114160, width:300, height:250});
}})(window,jQuery);
var o = document.getElementById('crt-820507849');
if ("undefined"!=typeof Criteo) {
var p = o.parentNode;
p.style.setProperty('display', 'inline-block', 'important');
o.style.setProperty('display', 'block', 'important');
Criteo.DisplayAcceptableAdIfAdblocked({zoneid:388248,containerid:"crt-820507849",collapseContainerIfNotAdblocked:true,"callifnotadblocked": function () {var o = document.getElementById('crt-820507849'); o.style.setProperty('display','none','important');o.style.setProperty('visbility','hidden','important'); } });
} else {
o.style.setProperty('display', 'none', 'important');
o.style.setProperty('visibility', 'hidden', 'important');
}
var o = document.getElementById('crt-1507690540');
if ("undefined"!=typeof Criteo) {
var p = o.parentNode;
p.style.setProperty('display', 'inline-block', 'important');
o.style.setProperty('display', 'block', 'important');
Criteo.DisplayAcceptableAdIfAdblocked({zoneid:837497,containerid:"crt-1507690540",collapseContainerIfNotAdblocked:true,"callifnotadblocked": function () {var o = document.getElementById('crt-1507690540'); o.style.setProperty('display','none','important');o.style.setProperty('visbility','hidden','important'); } });
} else {
o.style.setProperty('display', 'none', 'important');
o.style.setProperty('visibility', 'hidden', 'important');
}