Case-Study 1: Read the following passage and answer the questions given below.

An Apache helicopter of the enemy is flying along the curve given by y=x²+7. A soldier, placed at (3, 7) want to sheet down the helicopter when it is nearest to him.

(i)  If P(x₁, y₁) be the position of helicopter on curve y=x²+7, then find distance D from P to soldier place at (3, 7).

(ii) Find the critical point such that distance is minimum.

(iii) Verify by second derivative test that distance is minimum at (1, 8).

                                            OR

        Find the minimum distance between soldier and helicopter.

Given that P(x₁, y₁) is on curve y=x²+7 $\Rightarrow$.

      Let find the distance from P(x₁, $x_1^2+7$) and (3, 7) 

$$\begin{gathered} \mathrm{D}=\sqrt{{\left(x_1-3\right)}^2+{\left(x_1^2+7-7\right)}^2} \\ \Rightarrow \sqrt{{\left(x_1-3\right)}^2+{\left(x_1^2\right)}^2} \\ \Rightarrow D=\sqrt{x_1^4+x_1^2-6x_1+9} \end{gathered}$$

(ii) For minimum distance,

$\mathrm{D}=\sqrt{x_1^4+x_1^2-6x_1+9}$

${\mathrm{D}}^{\mathrm{\text{'}}}=x_1^4+x_1^2-6x_1+9$

$\frac{d{D}^{\mathrm{\text{'}}}}{dx}=4x_1^3+2x_1-6=0$

$\frac{{\mathrm{dD}}^{\mathrm{\text{'}}}}{\mathrm{dx}}=2{\mathrm{x}}_1^3+{\mathrm{x}}_1-3=0$

$\Rightarrow \left({\mathrm{x}}_1-1\right)\left(2{\mathrm{x}}_1^2+2{\mathrm{x}}_1+3\right)=0$

${\mathrm{x}}_1=1$ and $2x_1^2+2{\mathrm{x}}_1+3=0$ gives no real roots

The critical point is $\left(\mathrm{1,8}\right)$,

$\frac{d{D}^{\mathrm{\text{'}}}}{dx}=4x_1^3+2x_1-6$

$\frac{{\mathrm{d}}^2{\mathrm{D}}^{\mathrm{\text{'}}}}{{\mathrm{dx}}^2}=12{\mathrm{x}}_1^2+2$

${\left.\frac{{\mathrm{d}}^2{\mathrm{D}}^{\mathrm{\text{'}}}}{{\mathrm{dx}}^2}\right]}_{{\mathrm{x}}_1=1}=12+2=14>0$

Hence distance is minimum at $\left(\mathrm{1,8}\right).$

OR

For the minimum distance between soldier and helicopter at $\left(\mathrm{1,8}\right).$, it follows

$\begin{array}{rr} & D=\sqrt{x_1^4+x_1^2-6x_1+9}\\ & D=\sqrt{1+1-6+9}=\sqrt{5}\text{ units }\end{array}$

Therefore, (i) If $\mathrm{P}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ be the position of helicopter on curve $y={\mathrm{x}}^2+7$ , then the distance D from P to soldier place at $\left(\mathrm{3,7}\right)$ is $D=\sqrt{x_1^4+x_1^2-6x_1+9}$ 

(ii) The critical point such that distance is minimum is $\left(\mathrm{1,8}\right)$,

(iii) Proven that by second derivative test that distance is minimum at $\left(\mathrm{1,8}\right)$ the minimum distance between soldier and helicopter is $\sqrt{5}$ units.