A jet of an enemy is flying along the curve y = x2 + 2. A soldier is placed at the point (3, 2). What is the nearest distance between the soldier and the jet?
Let (x, y) be any point on y = x2 + 2 at which jet is at a particular moment.
∴ jet is at (x, x2 + 2)
Let d be the distance between the jet at (x, x2 + 2) and solider at (3, 2).
∴ d2 = (x – 3)2 + [ (x2 + 2) – 2]2 = (x – 3)2 + (x2)2
∴ d2 = x4 + x2 – 6 x + 9
Let f (x) = d2 = x4 + x2 – 6 x + 9
f ' (x) = 4 x3 + 2 x – 6 = 2 (2 x3 + x – 3) = 2 (x – 1) (2 x2 + 2 x + 3)
f ' (x) = 0 ⇒ 2 (x – 1) (2 x2 + 2 x + 3) = 0
⇒ x = 1 as we reject imaginary values of x.
f ' ' (x) = 12 x2 + 2
At x = 1, f ' ' (x) = 12 + 2 = 14 > 0 ⇒ f (x) has a local minimum at x = 1
But x = 1 is only extreme point
∴ f (x) is minimum at x = 1
∴ nearest distance = d at x = 1
