A.M.–G.M.–H.M. Inequality

Transcript

0:00Introduction

Hello everyone! Let's start with a small puzzle. Two positive numbers, x and y, add up to six. How big can x squared times y get? Many students split six equally, three and three. Three squared times three is twenty seven. So, is twenty seven really the biggest? Hmm, not so fast! There is a better pair, and one beautiful inequality will find it in a few lines. Questions like this come up often in JEE, so let's begin!

0:32The inequality of means

So, take n numbers, all of them positive, okay? We write them a one, a two, up to a n: the letter a with a small number below it. The three dots mean and so on. Keep that word, positive, in mind. It will matter today. Now, the means. The arithmetic mean, the A M: add the n numbers, and divide by n. The geometric mean, the G M: multiply them, and take the n th root. That is the small n sitting on the root sign. With me? And the harmonic mean, the H M: n over the sum of the reciprocals, one over a one, plus one over a two, up to one over a n. Right? Now my question. For the same positive numbers, which mean is the biggest, and which is the smallest? Yes, exactly! The A M is the biggest, and the H M is the smallest. This sign, greater than with a line below it, is read greater than or equal to. So, A M is greater than or equal to G M, which is greater than or equal to H M, clear? But when do all three means become the same number? That is correct, it happens when all the numbers are equal. Exactly then, and only then. This tells us where a greatest or least value is actually reached. Got it?

1:47Two numbers: A, G, H and G squared = AH

Now, two positive numbers, a and b. Capital A is a plus b, over two, and capital G is root a b. Capital H comes from the formula with n equal to two, which simplifies to two a b over, a plus b. Okay? Here is a lovely thing. Multiply capital A by capital H, and what do you get? Just a b! The a plus b cancels, the two cancels, and we are left with a b. The dot in the middle just means times. And a b is capital G squared. So capital G squared equals capital A times capital H, for any two positive numbers. Right? Quick check from the course. Two numbers have A M twenty five and H M nine. What is their G M? Root of twenty five times nine, that is, root two hundred and twenty five: fifteen! Perfect! And fifteen sits between nine and twenty five, just as the inequality says. With me?

2:43Using A.M. ≥ G.M.: two rules and a check

Alright, now the real use: greatest and least values. Rule one. For the least value of a sum, make the product of the terms constant. Then the G M is fixed, and the sum can never go below n times it. Okay? Rule two is the mirror image. For the greatest value of a product, make the sum constant. Then the A M is fixed, and the G M can never go above it. Clear? And now, the trap. This works only when all the terms are positive. And the bound is the answer only if equality can really happen, that is, if the terms can all be equal. Forget either check, and you get a wrong answer!

3:26Weighted AM-GM: greatest value of x^p y^q z^r

Now, a harder product. The greatest value of x to the power p, times y to the power q, times z to the power r. The small letters up top are powers, and here r is just a power, not a common ratio, okay? And x plus y plus z equals a fixed number k, all three positive. So, why not apply the A M and the G M to x, y and z directly? Because that only gives x y z, and we need x to the power p! So break x into p equal pieces, each x over p. Same for y, into q pieces, and z, into r pieces. The sum is still k, and now the product of the pieces has the right powers. Got it? And the maximum comes when all the pieces are equal. So x over p equals y over q equals z over r, equals k over, p plus q plus r. Toppers jump straight to this line. With me?

4:20Example 1: least value of x + 4/x for positive x

Now, Example one, an easy one. Find the least value of x plus four over x, for x greater than zero. Which rule fits? Pause the video, and try it first. Done? Okay, let's check. It is a sum, so we want a constant product. And x times four over x is just four. Constant! Both terms are positive, so the sum is at least two root four, that is, four. Right? But wait. Is four really reached, or is it just a floor we never touch? Equality needs the two terms equal: x equals four over x. The double arrow means it follows that. So x squared is four, and as x is positive, x equals two. Two plus two is four. Reached! Clear? So the least value is four. But why did the question say x greater than zero? What happens at x equals minus two? Minus two, plus four over minus two, that is minus four. Below four! Negative terms break the inequality. Always check positive first. Okay?

5:26Example 2: greatest value of x^2 y when x + y = 6

Now, Example two. Remember our puzzle from the start? Let's finish it. Find the greatest value of x squared times y, if x plus y equals six, with x and y positive. Pause the video and try it. Hint: the powers are not equal. Alright, it is a product with a fixed sum. So split equally, three and three, and get twenty seven, right? No, no, careful! That is the trap. In x squared y, x appears twice, but y only once. So split x into two halves: x over two, x over two, and then y. Three pieces, still adding up to six. Got it? Now apply the A M and the G M to these three positive pieces. Six is at least three times the cube root of their product. The small three on the root sign means cube root. And the product is x squared y over four. With me? Divide by three: two is at least that cube root. Cube both sides: eight is at least x squared y over four. So x squared y is at most thirty two. This sign, less than with a line below it, is read less than or equal to. Clear? But is thirty two reached? When are the three pieces equal? Each piece is six over three, that is, two. So x over two is two, which gives x equals four, and y equals two. Check: four squared is sixteen, times two is thirty two. Reached! Exactly the weighted rule. Right? So the greatest value is thirty two, not twenty seven. Puzzle solved! The equal split was the trap, because the powers were not equal. See how neat that was?

7:08Quick revision: fix the product or the sum, check equality

So, quick revision. For positive numbers, A M is at least G M, which is at least H M, with equality only when all the numbers are equal. For two numbers, capital G squared equals capital A times capital H, okay? Least sum: fix the product. Greatest product: fix the sum, and split by the powers. And always check: positive terms, and equality really possible. Now go and try the practice problems on this page. All the best!