Explanation:

To determine the maximum speed of the particle, we need to analyze the relationship between acceleration and velocity.

Relationship between acceleration and velocity:
The acceleration of a particle is the rate of change of its velocity with respect to time. Mathematically, it can be represented as:
a = dv/dt

Integrating both sides of the equation with respect to time, we get:
∫a dt = ∫dv

Simplifying the equation, we have:
∫a dt = v

This equation represents the relationship between acceleration and velocity.

Integration of acceleration:
Given that the acceleration of the particle varies with time, we need to integrate the acceleration function to find the velocity function.

If the acceleration at time t=0 is 5 m/s^2, we can represent the acceleration function as:
a(t) = 5t

To find the velocity function, we integrate the acceleration function with respect to time:
∫5t dt = 5∫t dt
∫5t dt = 5(t^2/2)
∫5t dt = (5/2)t^2

Therefore, the velocity function is:
v(t) = (5/2)t^2 + C

Determining the constant of integration:
To determine the constant of integration (C), we use the initial condition that the particle starts from rest, which means the initial velocity (v(0)) is 0.

Plugging in the values, we have:
v(0) = (5/2)(0^2) + C
0 = 0 + C
C = 0

Therefore, the velocity function becomes:
v(t) = (5/2)t^2

Maximum speed:
The maximum speed of the particle corresponds to the maximum value of the velocity function.

To find the maximum value, we take the derivative of the velocity function with respect to time and set it equal to zero:
dv/dt = 0

Differentiating the velocity function, we get:
dv/dt = (5/2) * 2t
dv/dt = 5t

Setting this equal to zero, we have:
5t = 0

This equation tells us that the velocity is zero at t = 0.

Conclusion:
Since the particle starts from rest, it will have a maximum speed of zero.

It may be infinity