Constraints on Acceleration Given Endpoints of Motion

[Kakashi24142]

Homework Statement

A particle of mass M moves on the X-axis as follows: It starts from rest at t = 0 from the point x = 0 and comes to rest at t = 1 at the point x = 1. No other information is available about it smotion at intermediate times (0 < t < 1). If α denotes the instantaneous acceleration of the particle, then prove that |α| must be ≥ 4 at some point in its path.

Homework Equations

$$\int_0^1{v(t)\,dt} = x(1) - x(0) = 1\\ \int_0^1{a(t)\,dt} = v(1) - v(0) = 0\\$$

The Attempt at a Solution

If constant accelerations of equal magnitudes are assumed for the period of acceleration and the period of deceleration, one obtains that the acceleration is 4 for the first 1/2 second and -4 for the last 1/2 second. I can see why the statement is true given these calculations, but could someone suggest a more rigorous way to prove this?

 

[mfb]

If |α| <=4, what is the quickest way to reach x=1? You can use symmetry to consider the acceleration part only, if you like.
If |α| <4, can you still have the same time?