Given that an apache helicopter of enemy is flying along the curve given by = x² + 7 

A soldier placed at (3,7) wants to shoot down the helicopter when it is nearest to him.

Now, the distance of the point from the soldier is S = √[ (x-3)² + (y-7)² ]

Since the point lies on the curve y = x² + 7 ..............1

     S = √[ (x-3)² + (x² + 7 - 7)² ]

=> S = √(x⁴ + x² - 6x + 9)

When the distance is maximum/minimum,

dS/dx = 0

(4x³ + 2x - 6) / √(x⁴ + x² - 6x + 9) = 0

=> 4x³ + 2x - 6 = 0

=> 2x³ + x -3 = 0

=> (x-1)*(2x² + 2x + 3) = 0

The solutions to the above equation are

x = 1, -0.5 ± i*0.5√5

Since the solution cannot have any complex roots.

Hence, x = 1 is the abscissa of the nearest point to the soldier.

From equation 1, we get

y = 1² + 7

=> y = 8

So, the nearest point is (1,8)

Now, nearest distance S = √[ (1-3)² + (8-7)² ]

=> S = √[ (-2)² + (1)² ]

=> S = √[4 + 1]

=> S = √5 units