What Is the Exact Value of x That Minimizes y for Positive x?

[recoil33]

I have a function which is:

y = 27x² + 2100/x
dy/dx = 54x - 2100/x²

Critical points are for what x values, is y undefined or equal to 0?

Therefore, in this case i can only see the value of 0, which would cause the function to be undefined. Due to x² being on the denominator.

EDIT:

0 = 54x - 2100/x²
0 = 54x³ - 2100
((-2100)/(54))^(1/3) = x

x = - 3.38
Although I'm looking for the amount of positive values of x, which y = 0.
So therefore, 1?

Question: "How many critical numbers does y have, for positive x?"
Solved, 1.

Which solved my next question, it's a local minimum.
______________________________________________________________________________

Give the exact value of the x that minimises y for positive x.
Could someone explain what i am looking for? I don't really understand.

I think.

P' = 54x - 2100/x²

Once x hits -3.38 it minises y?
Which is in decimal, but i could convert to a whole number?


Thank you.

 

[tiny-tim]

hi recoil33!

Give the exact value of the x that minimises y for positive x.
Could someone explain what i am looking for? I don't really understand.

they're asking for the value of x that you already found

Once x hits -3.38 it minises y?
Which is in decimal, but i could convert to a whole number?

i don't understand what they mean by "exact" …

unless they mean to keep it as (2100/54)^1/3, = (1050)^1/3/3

 

[HallsofIvy]

I have a function which is:

y = 27x² + 2100/x
dy/dx = 54x - 2100/x²

Critical points are for what x values, is y undefined or equal to 0?

Therefore, in this case i can only see the value of 0, which would cause the function to be undefined. Due to x² being on the denominator.

EDIT:

0 = 54x - 2100/x²
0 = 54x³ - 2100
((-2100)/(54))^(1/3) = x

You've misplaced the sign here. It should be x= (+2100/54)^1/3

x = - 3.38
Although I'm looking for the amount of positive values of x, which y = 0.
So therefore, 1?

Question: "How many critical numbers does y have, for positive x?"
Solved, 1.

Which solved my next question, it's a local minimum.
______________________________________________________________________________

Give the exact value of the x that minimises y for positive x.
Could someone explain what i am looking for? I don't really understand.

I think.

P' = 54x - 2100/x²

Once x hits -3.38 it minises y?
Which is in decimal, but i could convert to a whole number?


Thank you.

First, again, when you set y'= 0, 0= 54x- 2100/x² and add 2100/x² to both sides it becomes 2100/x²= 54x or x³= +2100/54 so your sign was wrong- this value of x is positive. Of course, you can reduce that fraction slightly: 2100/54= 1050/27 so the exact value of x is
x= (1050/27)^1/3= 1050[/sup]1/3[/sup]/3 as tiny-tim suggested.