1. Concept used: Escape energy

To send a body from the Earth's surface to infinity with just zero speed at infinity, the required initial kinetic energy must equal the increase in gravitational potential energy.

2. Gravitational potential energy at Earth's surface

The gravitational potential energy of a mass $m$ at distance $R$ from the center of Earth is

$U(R) = -\frac{GMm}{R}$

At infinity,

$U(\infty) = 0$

So the increase in potential energy needed is

$\Delta U = U(\infty) - U(R) = 0 - \left(-\frac{GMm}{R}\right) = \frac{GMm}{R}$

Thus the minimum kinetic energy required is

$K = \frac{GMm}{R}$

3. Use the relation between $g$ and $G,M,R$

We know

$g = \frac{GM}{R^2}$

So,

$GM = gR^2$

Substitute into the kinetic energy expression:

$K = \frac{GMm}{R} = \frac{gR^2 m}{R} = mgR$

4. Match with options

$K = mgR$

So the correct option is:

C: $mgR$

5. Verification with stored answer

Stored correct answer: C

Derived answer: C

They match.