Question

Find two consecutive natural numbers if the sum of the larger number and four times the reciprocal of the smaller number is a minimum.

Answer 1

can somone help me set it up ?? kinda confused what the question is saying

Answer 2

Try naming one of your variables and writing the others in terms of that one. I.e., let the smaller of the two consecutive numbers be n. Then the larger one is n+1 and their sum is
\[n+(n+1)=2n+1.\]

Answer 3

what do u get for the solution ?

Answer 4

Nothing, I wasn't actually planning on solving it. Sorry for the confusion!

Answer 5

do u get 2 and 3?

Answer 6

No, I actually got something else. I'm also curious what course this is for since I ended up differentiating the starting expression to find the solution.

Answer 7

calc 1

Answer 8

i think u have the main equation wrong

Answer 9

Oh, nevermind, I did get 2 and 3 as a local minimum and -2 and -1 as another local minimum. I didn't bother checking which is the absolute min.

Answer 10

i dont see any resciprocal

Answer 11

The equation should be
\[f(n)=n+1+\frac{ 4 }{ n },\]
Right?

Answer 12

yea

Answer 13

Oh, natural numbers. Great, looks like you're done.

Answer 14

can u help me with one more

Answer 15

wait could u exlain me why the equation should be like that? ?

Answer 16

Sure. If n is the smaller number, then \[n+1\] is the next one and \[\frac{4}{n}\] is four times the reciprocal of the smaller one. The equation I gave is just the sum of those two expressions.

Answer 17

how does n+1 become a smaller number ?

Answer 18

What are you referring to? The smaller number is n in my setup.

Answer 19

A possible solution using Mathematica is attached.

Answer 20

Thank you for the medal.

Answer 21

why did u add +1 after the n ?

Answer 22

Because it's the number that's one more than n. That's what "consecutive" means.

Answer 23

Review the English in the problem statement.
Let n be the larger number.
The smaller number n-1, is in the fraction.
n-1 and n are two consecutive numbers.

Answer 24

I guess that n could be defined as the smaller number and n+1 be the larger one.
I chose to encode the problem statement as it was written.

Answer 25

i have another question

Answer 26

Find two consecutive natural numbers if the sum of the larger number and four times the reciprocal of the smaller number is a minimum.

Answer 27

in this question i got 10 and -6 for my solition but i am wrong and idk what i did wrong ?

Answer 28

ok so for S i substitued x=4-y , but in the solution they sunbitied y=4-x and that is the only difference

Answer 29

Wait, is this for the same question?

Answer 30

no

Answer 31

@slaw

Answer 32

actaully hold on it is

Answer 33

this is question i wanted to ask , The sum of two positive numbers is 4. If the sum of their cubes is a minimum, what are these numbers?

Answer 34

Your method is fine because the equation is symmetric in x and y. That is, you can replace every x with a y and every y with an x to arrive at the same expression.

Answer 35

but does it matter ???

Answer 36

beacuse i see no differenec

Answer 37

If something went wrong, it wasn't that step.