00:00Given that x, y, and z are positive real numbers and x plus 2 times y plus z equals 12,
00:05The question asks for the greatest possible value of x times y squared times z times.
00:10Now, friends, can I write the x plus 2y here as x plus y plus z?
00:17I write.
00:18So, after I take it apart, when I actually divide it into four parts...
00:27What would I have gained?
00:283 is the result, which is the 4 numbers x, y, y, and z that I consider as 4 numbers.
00:35Doesn't it have an arithmetic mean?
00:37It is possible.
00:38I received 4 numbers.
00:40So, it turns out that x, y, y, and z are the arithmetic mean of the numbers.
00:52So we found the arithmetic mean.
00:54What am I going to do now, guys?
00:56We have x times y squared times z, right?
00:58This is the greatest value that is desired.
01:00How will we find this?
01:01This is also from the geometric mean, folks.
01:03How was the geometric mean?
01:05This is the geometric mean of these 4 numbers, sorry.
01:08Since there are four numbers, I'm taking it as the fourth root of x times y times y times z.
01:14And that equals, I say, my friends.
01:18Let's leave this aside for now before we move on to that.
01:22This is the geometric mean, folks.
01:24What is this?
01:24It is the geometric mean of the numbers x, y, y, and z.
01:32And I have the arithmetic mean of these four numbers, which are the same numbers.
01:35It also has a geometric mean.
01:36We had a rule.
01:38What was it, guys?
01:38The arithmetic mean of these 4 numbers is greater than or equal to.
01:42What's up, guys?
01:43What are geometric mean and greater than or equal to, friends?
01:47In its harmonic mean.
01:49Where will we use this time, guys?
01:52Of course, since there are two sets of data, we'll all use that side.
01:55What was our arithmetic mean, guys?
01:58I say 3 is greater than or equal to.
02:00What was our geometric mean, guys?
02:03I wrote that to show you the place.
02:04I can write it like this.
02:054, x times y squared times z.
02:08I did it again like that.
02:09Okay guys, what do I do to get rid of this fourth-degree root in my inequality?
02:14I'll get a score above 4 on every side.
02:18Will my inequality change direction?
02:19It won't change anything because it's positive.
02:21At the same time, these numbers are positive.
02:22So, 4, 4 roots are gone.
02:24What does x times y squared times z mean, guys?
02:29Less than or equal to.
02:30It should have been 81.
02:32It's the 4th power of 3.
02:33So that's the greatest value it can get.
02:39It was 81.
02:40Our answer is Ceyhan.
02:41See you.
02:422' Lorenz.
02:42Olnes.
02:425'e
02:425'e
02:427'e
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