Refer to Example 30, if the time of journey is minimum, find the value of v in terms of given quantities and also the minimum time.

Explanation

`t=d/v+v/2(1/alpha+1/beta)=dv^(-1)+v/2(1/alpha+1/beta)`
for t to be minimum
`(dt)/(dv)=d(-1)v^(-2)+1/2(1/alpha+1/beta)=0`
`d/v^(2)=(alpha+beta)/(2alphabeta)`
`v=sqrt((2alphabetad)/(alpha+beta))`
`t_(min)=d/sqrt((2alphabetad)/(alpha+beta))+1/2sqrt((2alphabetad)/(alpha+beta)).((alpha+beta)/(2alphabeta))`
`=sqrt(((alpha+beta)d)/ (2alphabeta))+=sqrt(((alpha+beta)d)/ (2alphabeta))`
`sqrt((2(alpha+beta)d)/(alphabeta))`