For $x \in \mathbb{R}$, the maximum value of $\sqrt{x^4 - 3x^2 - 6x + 13} - \sqrt{x^4 - x^2 + 1}$ is:
Step-by-Step Solution
Key Concept: Geometric interpretation transforms an algebraic expression into a distance problem where the maximum difference equals the distance between two fixed points.
The expression $\sqrt{(x-3)^2 + (x^2-2)^2} - \sqrt{x^2 + (x^2-1)^2}$ represents the difference of distances from point $P(x, x^2)$ on the parabola $y = x^2$ to points $A(3, 2)$ and $B(0, 1)$. By the triangle inequality, $(PA - PB)_{\max} = AB$.
Correct Answer: 2