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An observer whose least distance of dist...

An observer whose least distance of distinct vision is'd' views the his own face in a convex mirror of radius of curvature 'r'.Prove that maginification produced can not exceed `(r )/(d+sqrt(d^(2)+r^(2)) `

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`m=(f)/(f-u) rArrm=(f)/(f+x)`
`m=(r//2)/(r//2+x) rArr m=(R )/(r+2x)………(1)`
from mirror formula
`(1)/(v)+(1)/(u)=(1)/(f) rArr (1)/((d-x))+(1)/(-x)=(1)/(f) rArr(1)/(d-x)-(1)/(x)=(2)/(r)`
`(+x-(d-x))/(x(d-x))=(2)/(r) rArr (2x-d)r=2x(d-x)`
` 2rx-rd=2xd-2x^(2) rArr 2x^(2)+2(r-d)x-rd=0`
`2x^(2)+2(r-d)x-rd=0 rArr x=(-2(r-d)+-sqrt(4(r-d)^(2)+8rd))/(4)`
` x=-2(d-r)+-sqrt(4(r^(2)+d^(2)))/(4) rArr x=(d-r)+-sqrt(r^(2)+d^(2))/(2)`
`2x=(d-r)+-sqrt(r^(2)+d^(2))`
Putting the value of x in equation (1)
`m=(r)/(r+[(d-r)+-sqrt(r^(2)+d^(2))]`
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