Question

find two positive numbers whose product is 800 and the sum of the first plus twice the second is a minimum?

Answer 1

sounds like a lagrange multiplier problem

Answer 2

ab=800

Min=a+2b

hmm

Answer 3

f(a,b,L) = a+2b+L(ab-800) maybe, then partial it out

Answer 4

still dont get it :(

Answer 5

fa = 1+Lb = 0
b = -1/L

fb = 2+La = 0
a = -2/L

fL = ab-800 = 0 ; sub in a and b to solve for L

(-1/L * -2/L) - 800 = 0

2/L^2 - 800 = 0

(2 - 800L^2)/L^2 = 0

2 - 800L^2 = 0 ; when L^2=2/800

Answer 6

its about derivatives of exponential fuction.

Answer 7

it is? well, lets see if this works out cause I need to practice my Lagrange multipliers anyhoos :)

Answer 8

what is L?

Answer 9

L^2=2/800

L = -sqrt(2/800)

b = 1/sqrt(2/800) = 20
a = 2/sqrt(2/800) = 40

maybe

Answer 10

i mean L = minimum?

Answer 11

L is a Lagrange multiplier; which if you havent learnt it yet, you will later on

Answer 12

Its a way to compute a system of equations that have constraints

Answer 13

o... i haven't learned that one yet.. hahah is there any other way solving this problem by either using Derivatives of Exponential Fuction or Derivatives of Log?

Answer 14

i dont think there is a way to do it with exp or logs; but its been awhile so I could just be forgetting stuff

Answer 15

id solve it as a system of equations and derive them to see where the minimum is

Answer 16

ab=800; solve for say a=800/b

f(a,b)=a+2b; plug in our a value

800/b + 2b ; and derive

-800/b^2 + 2 i think is the derivative

Answer 17

when that equals 0 we got critical points

Answer 18

-800+2b^2 = 0 when b^2 = 800/2

b = sqrt(400) = 20

Answer 19

ab=800

a(20) = 800

a = 800/20

a = 40, jut like the lagrange worked out too yay!!

Answer 20

800/b + 2b ; and derive

-800/b^2 + 2 i think is the derivative

whats derive and how did you do this?

Answer 21

derive means to "take the derivative"

Answer 22

i assumed you knew that since you asked about taking the derivatives of exponential and log functions

Answer 23

we just learned that haha thank you !

Answer 24

but what is the meaning of the 'minimum?'

Answer 25

minimum, least, lowest point, can get lower than this, bottom of the barrel, etc ....

Answer 26

the smallest number

Answer 27

so in this question, it means the smaller critical point yes?

Answer 28

a+2b

40 + 2(20) = 80

80 is the smallest number you can get with the factors of 800

Answer 29

yes

Answer 30

the critical points themselves need to be dbl chked to make sure they are mins or maxs

Answer 31

got it! thank youuuu ;)