Use mean value theorem to show that for 0<x<y,


Expert Answers

An illustration of the letter 'A' in a speech bubbles

You need to rememeber what the mean value theorem tells such that:

`f'(c) = (f(b) - f(a))/(b - a)`

`f:[a,b]-gtR, c in [a,b]`

f continuous over [a,b]

Hence, you need to consider the function `f(z)=sqrt z` , f continuous over (x,y) such that:

`f'(z) = (sqrty - sqrtx)/(y-x)`

`(sqrt z)' = (sqrty - sqrtx)/(y-x) =gt 1/(2sqrtz) = (sqrty - sqrtx)/(y-x)`

`(y - x)/(2sqrt z) =(sqrty - sqrtx)`

You need to remember that `z in (x,y) =gt xltzlty`

`sqrtx lt sqrt z lt sqrt y`

`2sqrt x lt 2sqrt z lt 2sqrt y`

`1/(2sqrt x)gt 1/(2sqrt z)gt 1/(2sqrt y)`

Since `y gt x =gt y - x gt 0` , hence if you multiply by y - x inequality above yields:

`(y-x)/(2sqrt x) gt (y-x)/(2sqrt z) gt (y-x)/(2sqrt y)`

Notice that `(y - x)/(2sqrt z) = (sqrty - sqrtx)`  by Mean Value Theorem such that:

`(y-x)/(2sqrt x) gt (sqrty - sqrtx)`

Hence, by Mean Value Theorem yields that `(sqrty - sqrtx) lt (y-x)/(2sqrt x)`  under given conditions.

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial