Question

Find two positive real numbers whose sum is 100 and whose product is a maximum?

Answer 1

You can think of this as \[y = x*(100-x)\] which is an inverted parabola. What is the vertex?

Answer 2

Domain is (100, infinity)?

Answer 3

@kissy Let the numbers be x and y
we have a condition
\[x+y=100\]
Do you get this part?

Answer 4

absolutely

Answer 5

Have you studies calculus?

Answer 6

*studied

Answer 7

no, sorry. That's why i'm on here ):

Answer 8

I just want to know if you know differentiation and how to find the maxima and minima for a function. Of course I'll explain

Answer 9

Oh yes, I understand how to differentiate and how to find the extrema. I'm just having trouble understanding how to solve optimization problems

Answer 10

Ok, let's work on it

Answer 11

Let the product of x and y be P
so
\[P= x\times y\]
We know y=100-x
so
\[P=x\times (100-x)\]
Do you get this part?

Answer 12

yes

Answer 13

So we want to maximize P
Let's differentiate P with respect to x. Can you do that?

Answer 14

use chain rule to differentiate, right?

Answer 15

so P'(x)= 1(100-x)*(1)

Answer 16

It should be
\[P'(x)=1 \times (100-x)+x\times (0-1)\]

Answer 17

Shoot, forgot the x. Sorry

Answer 18

No problem, so we get
\[P'(x)=100-x-x\]
Can you find value of x is P'(x)=0

Answer 19

10?

Answer 20

\[0=100-x-x\]
\[0=100-2x\]
Can you try now?

Answer 21

Oh, I thought you meant what makes P'(x) 0 but you meant make P'(x) equal to 0. It is 50

Answer 22

Cool. I understood that. So there you go you got x , can you find y?

Answer 23

To find y you would make P'(0) right? ... as in you plug 0 into the x's of the original formula

Answer 24

We have the original condition
\[x+y=100\]
put x=50 and find y from this.

Answer 25

50 then.... where did you get x+y=100?

Answer 26

...nevermind. The questions states that the sum of x and y is 100 lol

Answer 27

Cool, so these are the two no.s which will make product the maximum

Answer 28

they are the CN's?

Answer 29

CN?

Answer 30

critical numbers

Answer 31

yes

Answer 32

do you understand?

Answer 33

Yes I do. I'm still a bit confused still though... like I understand what you did and i'm just still not understanding the rest of the steps

Answer 34

Just read it again, all of my posts. Let me know wherever you have doubt

Answer 35

Alright. I will look through again and try to figure it out on my own but if I still don't understand it I will let you know (: thanks

Answer 36

Cool

Answer 37

no calculus needed
the sum is 100, the product is a max, it is \(50\times 50\)

Answer 38

I still don't quite understand

Answer 39

ok you have two numbers whose sum is 100 right?

Answer 40

yes

Answer 41

so for example one could be 30 and the other 70
or one could be 25 and the other 75
or one could be 99 and the other 1
or even one could be 100 and the other one 0
in other words, you have two number \(x\) and \(y\) where \(x+y=100\) or if you prefer you have two numbers \(x\) and \(100-x\)

Answer 42

Oh, so the x and y that we just found are the two positive real numbers

Answer 43

yeah, just some real numbers
now here is the thing: you cannot tell then apart
by which i mean if you say one is \(x\) so the other is \(y=100-x\) or i say one is \(100-x\) and the other is \(x\) we both have the same thing, two numbers whose sum is 100

Answer 44

since you cannot distinguish between the numbers, \(x\) and \(y\) or \(x\) and \(100-x\), i.e. since they are interchangeable, the procedure for finding the maximum product cannot favor one number over the other
you say \(x+y=100\) and i say \(y+x=100\) we are both saying the same thing

Answer 45

so the maximum must be when the two numbers are identical, because we cannot tell them apart

Answer 46

you can check it with numbers and see what i mean
you say the first number is \(30\) so the second number must be \(70\) and \(30\times 70=2100\) now if i interchange them, we get \(70\times 30=2100\) the same answer exactly
now they will be biggest if we make both numbers the same, i.e. if \(x=y=50\) and \(50\times 50=2500\) is the biggest

Answer 47

Ok, I understand. So it'd be 50x50 for max and the two positive real numbers are 50 and 50 because like you said about the identical thing

Answer 48

if you don't like my logic which requires only common sense, we can also do it using algebra

Answer 49

right, you cannot tell them apart, so this procedure of finding the max cannot favor one number over the other
but we can still use algebra if this will make your teacher happy

Answer 50

Alright, how would you do it the algebra way? (it's not required but i'd still like to understand it)

Answer 51

call one number \(x\) so the other number must be \(100-x\) and their product is \(x(100-x)=100x-x^2\)

Answer 52

the equation
\[y=100x-x^2\] is a parabola that opens down
the biggest it can be is at the vertex, and the first coordinate of the vertex of a parabola is \(-\frac{b}{2a}\) which in this case is \(-\frac{100}{2\times (-1)}=50\)

Answer 53

therefore the maximum the product can be is if \(x=50\) which of course means the other number is \(50\) as well, and \(50\times 50=2500\)

Answer 54

it is simple enough algebra, but common sense is even simpler
how could it be possible that the maximum of the product could anything other than the number you would get if the two numbers were equal? why would one number be favored over the other? in other words, since \(x+y=100\) is symmetric in \(x\) and \(y\) by which i mean \(x+y=100\) is exactly the same as \(y+x=100\) it must be the case that the max is if \(x=y\)

Answer 55

Thank you!