Question

If we have that x+2y=20 then the largest possible value for xy is what
Maximum product?

Answer 1

let the product be P = xy.
since x+2y=20, solve for one these variables (x or y).
then plug this expression into the product equation.
what u got now?

Answer 2

x=20-2y and y=(20-x/2)

Answer 3

one of 'em will suffice.. let's choose that first one, x=20-2y.
so plugging that in to the product equation, P = xy, we get:
P = (20-2y)y
or
P = 20y - 2y^2

understand up to this point?

Answer 4

yes

Answer 5

so you need to maximize this P = 20y-2y^2.
without doing calculus you know this is a parabola opening downward and the maximum product will be the vertex of this parabola. did you want to do calculus on this or just find the vertex?

Answer 6

calculus

Answer 7

so find P' = ???

Answer 8

what I m getting I don't think is right

Answer 9

that's ok. ill check it. what u got? P' = ????

Answer 10

p=4y

Answer 11

P' = 20 - 4y
this is the correct first derivative. i think you forgot to take the derivative of the first term, 20y.

Answer 12

oh yeah

Answer 13

now set P' = 0 and solve. what u got for y = ???

Answer 14

y=-5

Answer 15

i think it's positive 5. double check please.... yes/no?

Answer 16

yup in is sorry

Answer 17

wait no it is -5

Answer 18

when you have -20=-4y when you bring the 4 over it turns to positive so the -20/4=-5

Answer 19

-20=-4y is correct. so solving this equation by dividing both sides by -4, you get:

\(\large \frac{-20}{-4}=\frac{-4y}{-4} \)
\(\large 5=\frac{\cancel{-4}y}{\cancel{-4}} \)

Answer 20

so y=5 right?

Answer 21

ok yes

Answer 22

ok... so y=5 and we need the x. do you know how we can get x = ???

Answer 23

no

Answer 24

remember x = 20 - 2y ???

Answer 25

this was from your constraint equation x + 2y = 20.

Answer 26

x=10

Answer 27

yes...
so P = xy = ???

Answer 28

5(10)

Answer 29

so then p=50

Answer 30

ok... that's the answer BUT..

Answer 31

Maximum product is50

Answer 32

yes... BUT

Answer 33

but what?

Answer 34

how do you know that is the maximum?

Answer 35

in any optimization problem you must justify that that is either a maximum or minimum...

Answer 36

ok well maybe I was jumping the gun so p=5(10) so then what

Answer 37

no... 50 is correct... the maximum product is 50

Answer 38

ok are you sure

Answer 39

you just need to justify that this product is the maximum... and yes, i'm sure...

Answer 40

ok

Answer 41

how do you justify that this is the maximum?

Answer 42

by checkin it right?

Answer 43

that would take forever because the domain is all real numbers...
maybe you can use the second derivative test?
what's P'' =

Answer 44

it would be um not sure I can;t get it to come out

Answer 45

first derivative: P' = 20 - 4y
second derivatve: P'' = -4

notice P'' is ALWAYS negative.. which means P is concave up or down?

Answer 46

concave donw

Answer 47

yes.... since it's concave down, the critical number is a maximum.

this also verifies that in the beginning i told you this was a parabola opening downwards...
you did awesome with this probem... thanks for being a great student! :)

Answer 48

ok thaks