The maximum value of $\left|\sqrt{(x^2-2)^2+(x-3)^2} - \sqrt{(x^2+2)^2+x^2}\right|$ is
Step-by-Step Solution
Key Concept: Interpret the expressions as distances: $P=(x, x^2)$ is a point on the parabola $y=x^2$. The first radical is $|PA|$ where $A=(3,2)$, and the second is $|PB|$ where $B=(0,-2)$. Use $|PA-PB| \leq AB$.
Let $P=(x,x^2)$ on $y=x^2$. Distance $PA = \sqrt{(x^2-2)^2+(x-3)^2}$, $PB = \sqrt{(x^2+2)^2+x^2}$. By triangle inequality $|PA-PB| \leq AB = \sqrt{9+16} = 5$.
Correct Answer: (D) 5