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The rules regarding an Alchemist's bombs states:

On a direct hit, an alchemist's bomb inflicts 1d6 points of fire damage + additional damage equal to the alchemist's Intelligence modifier. The damage of an alchemist's bomb increases by 1d6 points at every odd-numbered alchemist level (this bonus damage is not multiplied on a critical hit or by using feats such as Vital Strike). Splash damage from an alchemist bomb is always equal to the bomb's minimum damage (so if the bomb would deal 2d6+4 points of fire damage on a direct hit, its splash damage would be 6 points of fire damage).

Now this means a 1st level Alchemist with an Int of 16 will do 1d6+3 damage. If he rolls a 2 for damage, he will do 5 points damage, with 4 points splash damage. On a crit, he will double his damage to 10 points, but nowhere can I find anything about critical damage from the splash effect. Usually splash weapons do not deal critical damage. Does this mean the bomb only deals its critical damage on a direct hit, and never on its splash damage?

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Only effects requiring attack rolls yield critical hits

The splash damage of an alchemist's bomb isn't increased on a critical hit because the alchemist didn't make an attack roll to hit the splashed creatures with the splash damage. The rules for Critical Hits say

When you make an attack roll and get a natural 20 (the d20 shows 20), you hit regardless of your target's Armor Class, and you have scored a "threat," meaning the hit might be a critical hit (or "crit"). To find out if it's a critical hit, you immediately make an attempt to "confirm" the critical hit—another attack roll with all the same modifiers as the attack roll you just made. If the confirmation roll also results in a hit against the target's AC, your original hit is a critical hit. (The critical roll just needs to hit to give you a crit, it doesn't need to come up 20 again.) If the confirmation roll is a miss, then your hit is just a regular hit.

A critical hit means that you roll your damage more than once, with all your usual bonuses, and add the rolls together. Unless otherwise specified, the threat range for a critical hit on an attack roll is 20, and the multiplier is ×2.

Emphasis mine. The rules for when a creature takes the action Throw Splash Weapon include

A hit deals direct hit damage to the target, and splash damage to all creatures within 5 feet of the target.

The alchemist made an attack roll to hit directly the target; the alchemist didn't make an attack roll to hit the creatures within 5 ft. of the target. Thus even on a critical hit the alchemist deals his normal splash damage (although he still deals extra damage to the target hit directly).

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