Calculus question (1 Viewer)

[cutemouse]

Hello,

How do I do this question? Thanks

Show x+(1/x) >_ 2 , for all x>0
(ie. x+(1/x) is greater than or equal to 2)

Thanks again!

 

[shady145]

you could draw a graph n show it on that

 

[cutemouse]

Is there any other way I could do this?

Thanks

 

[lolokay]

simplest/most common way is using the fact that (x-1)²>= 0

you could also use calculus

 

[Timothy.Siu]

Hello,

How do I do this question? Thanks

Show x+(1/x) >_ 2 , for all x>0
(ie. x+(1/x) is greater than or equal to 2)

Thanks again!

i.e. x+1/x-2 >= 0


[(x-1)^2]/x >=0

true for all x>0

you could also use induction but thats quite long lol

 

[lolokay]

since title says calculus question, i'll expand on that method a bit

basically, differentiate x + 1/x to find the stationary point, and then find the second derivative so you can test if it's a min/max/inflection (remembering x>0)

you'll find that it's a minimum, and occurs at x=1, which is when equality holds
.'. it is larger for all other positive values

 

[rokkuran-x]

i would have used induction but calculus works too

 

[akatsukisan]

no calculus or induction needed, not even a calculus question i.e.

(x^0.5-((1/x)^0.5))^2>0 for all x

Expand:

x-2+(1/x)>0
x+(1/x)>0

:. true for all x

but bad question - should be bigger than or equal to minus two i.e. for x=-1 or a domain for x should have been specified.