A jet of enemy is along the curve `y=x^2+2` and a soldier is placed at (3,2).Find the minimum distance between the jet and soldier.

Explanation

Let `P(x,y)` be the position of the jet and the soldier is placed at `A(3,2)`.
`=>AP=sqrt((x-3)^2+(y-2)^2).....(i)`
As `y=x^2+2`
`=>x^2=y-2.....(ii)`
`=>AP^2=(x-3)^2+x^4=z`(say)
`=>(dz)/(dx)=2(x-3)+4x^3`
`(d^2z)/(dx^2)=2+12x^2`
For local points of maxima/minima,
`(dz)/(dx)=0`
`=>2(x-3)+4x^3=0`
`=>x=1`
` [(d^2z)/(dx^2)]_(at x=1)=14 gt 0 `
`∴
z` is minimum when `x=1,y=1+2=3`

Also minimum distance `= sqrt((1-3)^2+(3-2)^2)=sqrt5` units.