find two positive numbers whose product is 100 and whose sum is minimum

find two positive numbers whose product is 100 and whose sum is minimum, calculus optimization
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[@volodymyrgandzhuk361]

Of all the rectangles with equal area, the one with the smallest perimeter is the square.

[@lucasmartiniano6915]

Wow

[@mastermafia160]

Also, of all the rectangles with equal perimeters, the one with largest area is the square

[@volodymyrgandzhuk361]

@@mastermafia160 he asked for minimum sum, not for maximum product

[@mastermafia160]

@@volodymyrgandzhuk361 chill i was just spitting facts

[@stlemur]

You can do this without calculus by rearranging the equation for S(x), recognizing the quadratic, and finding the vertex

[@willie333b]

How

[@stlemur]

@@willie333b S = x + 100/x, then x^2 - Sx + 100 = 0. Left hand side as a function of x describes a parabola with vertex at x = S/2 or S = 2x, implying x = y.

[@willie333b]

@@stlemur thanks
I forgot what I learned in junior high 😅

[@stevemonkey6666]

Of all of these problems the answer always is the smallest is a square. Why not prove why the smallest is a square?

[@officiallyaninja]

try it yourself. xy = A and minimize x+y, you'll get x=y

[@hamzahar8793]

set xy=a for some positive a, so y=a/x. x+y=x+a/x. we can find the minimum using am gm inequality, (x+a/x)/2>=sqrt(x*a/x). so in the end its x+a/x>=2sqrt(a), so the minimum is 2sqrt(a). make it a quadratic equation: x^2-2sqrt(a)x+a=0 which can simplify into (x-sqrt(a))^2=0 so the solution is x=sqrt(a).

[@prashanthkumar0]

We can do it with AM≥GM unequality
(a+b)/2 ≥ ✓ab (given ab=100)
=> a+b=20
a,b satisfy x²-20x+100 => a,b=10,10

[@GameJam230]

I've been obsessed with prime factorization recently, and have learned for a fact that the further away two factors are from the square root of the given number, the larger their sum is. this is because if you halve one factor, the other must be doubled for the result to stay equal, and the distance between X and 2X is always greater than X/2 and X, thus the sum of the two will increase because the amount gone up by one factor is greater than the amount gone down by the other.

[@kittynpaws6580]

Asians are so smart lol😮
Your mom must be proud of you now 😂❤

[@fanamatakecick97]

Are these all proofs that squares are the answer?

[@Sonny2009]

Think of it this way. When you have the derivate is 0 at that point, the function can be MAXIMUM or MINIMUM. So we try to see When the derivate is negative or derivative is positive. At those point, we know the function is increasing or decreasing(at this point we also start scratching our head badly)

[@MohammadIbrahim-sq1xn]

could be easily done by AM >= GM

[@WerewolfLord]

Now find two integers whose product is 100 and sum is minimum. 😈

[@Saksham_or_smth]

He did it already

[@Gold161803]

That would be -1 and -100 I guess. Extend to real numbers and you got nothing though

[@broytingaravsol]

mimimum

[@India1845-e5w]

Sir, how will be the minimum size of their square, just tell me, I have found out the answer.

[@mikefochtman7164]

While I didn't have a proof, it seemed to me intuitively that the answer has to be the square root of the product mentioned (in this case the square root of 100). As you move away from the square root, one number has to grow faster than the other number shrinks to maintain the same product. So the sum must always grow as you move away from the square root.

[@AlfonsoNeilJimenezCasallas]

So, if x*y = 100, x = 10 and y = 10, the minimum sum is x + y = 10 + 10 = 20

[@MrConverse]

This video was recorded years ago, am I right?

[@dfsgjlgsdklgjnmsidrg]

why is he doing preschooler math

[@volodymyrgandzhuk361]

Since when does preschool math involve calculus?