Math question (sort of)

We have:
X > 1, Y > 1 (X and Y are variables)
X + Y = Z (Z is a constant)
X * Y = C

Assumption: We will get max value of C when X = Y.

Questions:
Is the above assumption correct? If it is, does this whole thingy have some kind of a name or something (it feels like it does, but I can't remember what it might be called and it's bugging me something fierce)?

PoE application (or why this question was bothering me in the first place):
When you are considering whether to get %increased damage or +%crit multi on a max crit chance build (assuming equal values, like 40% increased or +40% crit multi), if the above assumption is correct, then you should get the one that has a lower total.

So if (total %increased + 100%) < (total %crit multi), then X% increased damage will give more damage, than +X% crit multi.

Here's hoping my english didn't make the whole thing too confusing to read :), and thanks in advance.

The purely mathematical assumptions are correct; there are two things you're failing to consider.

(1) the investment to get x% additional crit multiplier is different from the investment required to get x% increased damage. If it's twice as hard to get 300% extra crit multi than 300% increased damage, then trying to force the balance is foolish.
(2) sometimes, x% increased damage scales more because of the principle of "double dipping". For example, burn and poison builds. x% pure increased damage increases burn scaling and poison scaling on top of the base hit scaling. x% additional crit multiplier only increases the base hit scaling and not the burn or poison scaling, so you want it to be lower up to a certain point.

As for the name of the system, I don't remember.

My bad, forgot to mention that the PoE example only works in a vacuum, pretty much. Naturally, there's more things to consider when dealing with real character. And damage optimization on such a small scale is probably not the most important part for a crit build anyway :).

Thanks for confirming the math part.

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PaperRat wrote:

Is the above assumption correct? If it is, does this whole thingy have some kind of a name or something (it feels like it does, but I can't remember what it might be called and it's bugging me something fierce)?

Correct. Solvind the first equality for X gives X = Z - Y. Applying this to the second equality yields X * (Z - X) = C. Rearrange terms to get the polynomial -X*X + Z*X - C = 0. Since the coefficient of the quadratic term is negative, it has a maximum point. It so happens this maximum occurs at the point where the derivative is zero. In other words where -2*X + Z = 0. Solving for X gives X = Z/2. This means that also Y = Z/2 and X = Y. QED.

In order for this to hold in the case of increased damage and crit multiplier, you must also assume that crit chance is 100%. This is definitely possible so it doesn't invalidate your claim, but it's something to be aware of.

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databeaver wrote:

In order for this to hold in the case of increased damage and crit multiplier, you must also assume that crit chance is 100%. This is definitely possible so it doesn't invalidate your claim, but it's something to be aware of.

It's possible in the sense that there's a weapon that guarantees crits, but it's not possible for any other build to surpass 95%.

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It's possible in the sense that there's a weapon that guarantees crits, but it's not possible for any other build to surpass 95%.

My bad, I've never done a crit capped build so I forgot that. It's possible to reach the cap though.

The 95% cap means that for every 100% increased damage you should have 105% added crit multiplier (including the 50% you get for free).

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databeaver wrote:

In order for this to hold in the case of increased damage and crit multiplier, you must also assume that crit chance is 100%. This is definitely possible so it doesn't invalidate your claim, but it's something to be aware of.

It's possible in the sense that there's a weapon that guarantees crits, but it's not possible for any other build to surpass 95%.

Doesn't Assassin's Mark push the 95% chance over a hundred? Or is additional chance to crit essentially "wasted" on a crit capped build?

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PaperRat wrote:

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databeaver wrote:

In order for this to hold in the case of increased damage and crit multiplier, you must also assume that crit chance is 100%. This is definitely possible so it doesn't invalidate your claim, but it's something to be aware of.

It's possible in the sense that there's a weapon that guarantees crits, but it's not possible for any other build to surpass 95%.

Doesn't Assassin's Mark push the 95% chance over a hundred? Or is additional chance to crit essentially "wasted" on a crit capped build?

Yep Assassn's mark is an additive crit chance and can bring you to (and above maybe?) 100% crit chance. You also have to factor hit chance since crit is rolled twice.

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PaperRat wrote:

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databeaver wrote:

In order for this to hold in the case of increased damage and crit multiplier, you must also assume that crit chance is 100%. This is definitely possible so it doesn't invalidate your claim, but it's something to be aware of.

It's possible in the sense that there's a weapon that guarantees crits, but it's not possible for any other build to surpass 95%.

Doesn't Assassin's Mark push the 95% chance over a hundred? Or is additional chance to crit essentially "wasted" on a crit capped build?

Riiight. Good call, my mistake.

just factor in the weighted averages of the crits then