Final answer:
The force f = -αx²i may be conservative if it can be derived from a potential energy function and if the work done by the force is path-independent, but additional information or context is needed to conclusively answer the question.
Step-by-step explanation:
To determine if the force f = -αx²i is conservative, we consider whether the work done by this force is independent of the path taken between two points. A conservative force has a corresponding potential energy and satisfies the condition that the curl of the force is zero. In mathematical terms, this means if you take the partial derivatives dFy/dx and dFx/dy, and they are equal, then the force is conservative. Since the given force only has an x component, dFx/dy is zero, and we do not have a corresponding y-component of the force to find dFy/dx. Therefore, by our criteria for a conservative force, it is unclear with the given information whether the force is conservative. Typically, forces that depend only on position and are derived from a potential energy function are conservative, so without additional context or components of the force, we cannot definitively answer this question. However, if this force were the only one at work and if it could be derived from a potential energy function, it could be considered conservative.