A slightly mathematical answer:

The motion of planets around the sun traces out a conic section(ellipse). At the same time, the eccentricity of any body exhibiting central force motion has the value: $$\epsilon=\sqrt{1+\frac{2EL^2}{\mu C^2}}$$

where E=total energy of the body L=angular momentum(constant) of the body $\mu$=reduced mass(in the case of planetary motion this is just $M_{planet}$ C=central force motion constant(for gravitation $GM_s$)

for the path to be unbounded: we must have $\epsilon>=1$. For this the total energy must be $>=0$. The extra energy to be provided is $E_{\epsilon>=1}-E_{ellipse}$

Since we have to provide the minimum energy, we have to choose $\epsilon$ such that $E_{\epsilon>=1}$ is minimum.

This corresponds to $\epsilon=1$, or $E_{\epsilon>=1}=0$(parabolic path).

Hence the total energy at infinity must be 0.

Edit 1: $E_{ellipse}=-GMm/2a$ where M= Mass of sun 2a= major axis of ellipse m=mass of body

A slightly mathematical answer:

The motion of planets around the sun traces out a conic section  (ellipse). At the same time, the eccentricity of any body exhibiting central force motion has the value: $$\epsilon=\sqrt{1+\frac{2EL^2}{\mu C^2}},$$

where

• $E$ = total energy of the body
• $L$ = (constant) angular momentum of the body
• $\mu$ = reduced mass (in the case of planetary motion this is just $M_{\text{planet}}$
• $C$ = central force motion constant (for gravitation, $GM_s$).

For the path to be unbounded, we must have $\epsilon \geq 1$. 
For this, the total energy must be $\geq 0$. 
The extra energy to be provided is $E_{\large \epsilon \ \geq \ 1}-E_{\text{ellipse}}$.

Since we have to provide the minimum energy, we have to choose $\epsilon$ such that $E_{\large \epsilon \ \geq \ 1}$ is minimum.

This corresponds to $\epsilon=1$, or $E_{\large \epsilon \ \geq \ 1}=0$ (parabolic path).

Hence, the total energy at infinity must be $0$.

Edit 1: $$E_{\text{ellipse}}=-\frac{GMm}{2a},$$ where

• $M$ = Mass of the Sun
• $2a$ = major axis of ellipse
• $m$ = mass of the body.

A slightly mathematical answer:

The motion of planets around the sun traces out a conic section(ellipse). At the same time, the eccentricity of any body exhibiting central force motion has the value: $$\epsilon=\sqrt{1+\frac{2EL^2}{\mu C}}$$

where E=total energy of the body L=angular momentum(constant) of the body $\mu$=reduced mass(in the case of planetary motion this is just $M_{planet}$ C=central force motion constant(for gravitation $GM_s$)

for the path to be unbounded: we must have $\epsilon>=1$. For this the total energy must be $>=0$. The extra energy to be provided is $E_{\epsilon>=1}-E_{ellipse}$

Since we have to provide the minimum energy, we have to choose $\epsilon$ such that $E_{\epsilon>=1}$ is minimum.

This corresponds to $\epsilon=1$, or $E_{\epsilon>=1}=0$(parabolic path).

Hence the total energy at infinity must be 0.

Edit 1: $E_{ellipse}=-GMm/2a$ where M= Mass of sun 2a= major axis of ellipse m=mass of body

A slightly mathematical answer:

The motion of planets around the sun traces out a conic section(ellipse). At the same time, the eccentricity of any body exhibiting central force motion has the value: $$\epsilon=\sqrt{1+\frac{2EL^2}{\mu C^2}}$$

where E=total energy of the body L=angular momentum(constant) of the body $\mu$=reduced mass(in the case of planetary motion this is just $M_{planet}$ C=central force motion constant(for gravitation $GM_s$)

for the path to be unbounded: we must have $\epsilon>=1$. For this the total energy must be $>=0$. The extra energy to be provided is $E_{\epsilon>=1}-E_{ellipse}$

Since we have to provide the minimum energy, we have to choose $\epsilon$ such that $E_{\epsilon>=1}$ is minimum.

This corresponds to $\epsilon=1$, or $E_{\epsilon>=1}=0$(parabolic path).

Hence the total energy at infinity must be 0.

Edit 1: $E_{ellipse}=-GMm/2a$ where M= Mass of sun 2a= major axis of ellipse m=mass of body