The position of a particle moving along the x-axis is given by a twice-differentiable function x(t). If x(2) < 0, x'(2) < 0, and x''(2) <0, which of the following statements must be true about the particle at time t=2? a. The particle is moving toward the origin at a decreasing speed.
b. The particle is moving toward the origin at an increasing speed.
c. The particle is moving away from the origin at a decreasing speed.
d. The particle is moving away from the origin at an increasing speed.
e. The particle is moving away from the origin at a constant speed.

c. The particle is moving away from the origin at a decreasing speed.

Step-by-step explanation:

Since x(2) which represents the particles position is less than zero, that is, x(2) < 0, it shows that the particle is at the left of the origin at t = 2.

Also, x'(2) which represents the particle's speed at t = 2 is less than zero, that is, x(2) < 0, the particles speed is negative at t = 2.

Finally, since x"(2) which represents the particle's acceleration at t = 2 is less than zero, that is, x"(2) < 0, the particle's acceleration is negative at t = 2 and thus decreasing.

So, combining these three conditions together shows that the particle is moving away from the origin at a decreasing speed at time, t = 2.

astronomers is your final answer! : )