Hi there, I have a Jax Hybrid AP build a little different from yours but would really like to put some additions for you to research/accommodate on your build.

-I call this build "Frogger Jax" lols

-Masteries I go 15/15/0 taking both ap and ad penetration, skipping attack speed and taking spellsword. defense health, 2 points on hardiness juggernaut and relentless

-AD marks (early game, last hitting, trades, etc), Armour seals, MR/level glyphs, Movement speed Quints

-Core Items:

Boots of choice (depending on team comp)
Gunblade
Nashors
Gauntlet
Rod of ages

-Last item is optional and situational:

Cap
Abysmal
Randuims
Hourglass
Statikk
Liandrys
Void
Wits end
Any HP defensive item
Any Armour or MR item

-for level 18 your stats should look something like this (not including last item):

3K HP
150+ armour (before R)
100+ mr (Before R)
34% CDR (leap 4s, empower 2s, Counter 6s)
170+ AD
210+ AP
12% Lifesteal
20% Vamp

WARNING HUGE WALL OF TEXT, PLEASE READ IT ALL

I don't care that this is 8 months later when you said this, people need to stop thinking this and you need to not give out false information in a popular guide. Every 1 point of armor (or magic resist) makes your HP 1% more effective. If you don't understand that much, I will explain more.

If you have 1000 hp with 0 armor and the enemy does 1000 damage, you die in one hit.

Say you buy a thornmail, 100 armor. Your E(ffective) HP is now 2000, so you die in two hits.

Say you buy another thornmail, 200 armor. you EHP is 3000. That is 1000 more effective health for 100 more armor.

with 1000 hp, 100 armor = 1000 EHP

DOUBLING ARMOR != DOUBLE EHP
HOW DO YOU NOT SEE THAT FROM WHAT THE LAST GUY SAID.

Here, ill make it even easier if you STILL CAN'T SEE WHY THERE IS 0 DIMINISHING RETURNS

Say you have 1000 hp, 0 armor, enemy that does 1000 damage. you still die in one hit.

But now you get a warmongs, 1000 hp. well your EHP is now 2000, you die in two hits.

You get another warmongs, 1000 more hp. now your EHP is 3000, you die in three hits

You for some reason assume doubling armor (or mr) doubles your effective hp. Do you also think that getting two warmongs instead of one will double your overall hp? No? That is what I thought.

Easiest way to think of it is this: Think that when you buy armor or mr, you are actually buying invisible hp that is based off of your visible hp. 1 point of armor always increases the invisable hp by the same ammount every time. 100 armor always increases it the same as the last 100

Now that isn't to say you shouldn't buy hp, because more hp makes your resistances more valuable, while more resistance makes your hp more valuable. There is a balance between the two that will give you the most effective hp. Now please, just remove that section about diminishing returns, I immediately stop reading guides/discredit people who say that while I try to help them. If I can say one thing to everybody about diminishing returns, just ignore the % damage reduction as it is very, VERY misleading.

In the late game build section, you showed the percentages of armor for the amount of armor and deducted that armor becomes less effective after ~100-150 armor. This isn't true. Let me tell you something about effective health.
If you calculate the damage output, every point of armor or magic resistance is 1% of effective health.
Examples:
1) if a champion has 1000 health and 100 armor, this champion has 100% effective health, meaning that they will need to take 2000 damage (before armor calculations) to die.
2) if a champion has 1000 health and 300 armor, this champion has 300% effective, meaning that they will need to take 4000 damage (before armor calculations) to die.
So if a champion deals 500 damage per hit, then example one will take 4 hits to die.
Calculations: 500*(100/(100+100)) = 250 damage, and 1000/250=4, so yes, 4 hits to die.
With this same champion dealing 500 damage per hit, example two will take 8 hits to die.
Calculations: 500*(100/(100+300)) = 125 damage, and 1000/125=8, so 8 hits to die.
Now remember that armor penetration will affect this damage and reduce the amount of effective health someone has, but I just wanted to let you know that armor and magic resist do not become less effective as you buy more.