Identify the forces involved and the conditions for equilibrium.

To keep the box from falling, the upward static friction force ($F_{fr}$) must balance the downward gravitational force ($mg$).

$$F_{fr} = mg$$

The horizontal force ($F$) applied to the box creates a normal force ($F_N$) from the wall. Since the box is not moving horizontally, these forces are equal in magnitude.

$$F = F_N$$

The maximum static friction force is given by the formula:

$$F_{fr,max} = \mu_s F_N$$

For the box to not fall, the static friction force must be at least equal to the gravitational force. The minimum applied force $F$ corresponds to the situation where the static friction is at its maximum value.

We know $F_{fr} = mg$ and $F_N = F$. Substitute these into the maximum static friction formula:

$$mg = \mu_s F$$

Rearrange the equation to solve for $F$:

$$F = \frac{mg}{\mu_s}$$

Given $\mu_s = 0.40$ and $mg = 20 \text{ N}$:

$$F = \frac{20 \text{ N}}{0.40}$$ $$F = 50 \text{ N}$$

The calculated minimum force is $50 \text{ N}$, which corresponds to option (c).

The final answer is $\text{c) } 50 \text{ N}$.