00:00Let A be a positive number. The question asks for the smallest possible value of the expression 9A plus 18 divided by A.
00:06Now we know, friends, that we had a system of inequality, what was it?
00:10Isn't that the harmonic mean, guys, in the demetric mean of two numbers that are less than or equal to?
00:18What is less than or equal to?
00:20The arithmetic mean of the numbers, right guys?
00:24Yes, where am I going to use it here?
00:26If we consider these two numbers, then friends, shouldn't I be using this part?
00:32Okay, let's get here then.
00:35Let our numbers be 9A and 18 divided by A.
00:40And also, friends, I'm talking about our 9A plus our 18 divided by A plus the average.
00:52What are we going to do?
00:54We will collect them.
00:54I'll say division 2.
00:55So what does that equal, friends?
00:58I find it as 9A squared plus 18 divided by 2A.
01:022 times A.
01:03So what is this, guys?
01:05Arithmetic mean.
01:06Let's look at the geometric mean.
01:08What is that equal to?
01:10How many of us were there?
01:10There were two numbers, and they were 9A and 18 divided by A.
01:13Now, what will we achieve?
01:14What is 9A multiplied by 18 divided by A?
01:18We'll take its dark roots.
01:20Right, guys?
01:20Since it's a quadratic root and there are two numbers involved, taking their square roots gives us the geometric mean.
01:24I say that's equal to that too.
01:26Because 'A's are only used in multiplication and division.
01:289 will go outside, right guys?
01:31Yes.
01:32We find this place as 9 square root of 2.
01:37And what was ours, friends?
01:38Right here.
01:429A squared plus 18 divided by A, 2 times A, sorry.
01:47What was it, guys?
01:48The arithmetic mean was greater than or equal to the geometric mean.
01:519 square root of 2.
01:52Dear friends, it seems the conjugated form needs to be greater than 9 square root of 2.
01:58Or they were supposed to be equal.
02:00So now, what else do we need, friends?
02:03How did we find it?
02:04We were asked to find the sum of these two numbers, right?
02:07What is this, guys?
02:071/2 times the original amount.
02:08If we multiply each side by 2, will our uniqueness change?
02:15Constant.
02:16The 2s are just the password.
02:179A squared plus 18 divided by A is greater than or equal to.
02:21What's up, guys?
02:2318 square root of 2.
02:25So, the smallest possible value for the sum of our friends will either be 18 square root of 2 or greater than 10.
02:32In this case, the smallest possible value for our answer is 18 square root of 2.
02:35The answer is Adana.
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