TITLE: Find maximum of $|\sqrt{x^4-7x^2-4x+20}-\sqrt{x^4+9x^2+16}|$ without using calculus
QUESTION [7 upvotes]: Find maximum of $|\sqrt{x^4-7x^2-4x+20}-\sqrt{x^4+9x^2+16}|$ without using any calculus

I believe I have made most of the progress but am stuck on the last step
I rewrote the expression as- $$\left|\sqrt{(x^2-4)^2+(x-2)^2}-\sqrt{(x^2+4)^2+(x-0)^2}\right|$$
We notice that this is written in the format of the distance formula. In particular, this is the expression for the difference of distances from an arbitrary point $(x^2,x)$ and $(4,2)$,$(-4,0)$ respectively.

In this figure we have to find maximum of $|PA-PB|$
I could not think of how this could be maximised. Usually in questions like these $P$ somehow is collinear or lies on the reflection of some given point but nothing of that sort can happen here. I tried a few special points as well like $(4,-2),(0,0)$ etc.
Can someone help me with this?

REPLY [4 votes]: By the triangle inequality, for every point $P$,
$$||PA|-|PB||\leq |AB|$$
so the maximum is attained when $P=A$.