Minimum value of $\sqrt{x^4 + 3x^2 - 6x + 10} + \sqrt{x^4 - 5x^2 + 9}$ without using calculus?

Question

Hi mathematics stack exchange, what is the minimum value of $\sqrt{x^4 + 3x^2 - 6x + 10} + \sqrt{x^4 - 5x^2 + 9}$? I know how to solve this problem using calculus, you take a derivative, but I am wondering if there is an elementary method to find the minimum using precalculus methods.

Answer 1

Let $y = x^2$, notice

$$\begin{align} x^2 + (y-3)^2 & = x^2 + (x^2-3)^2 = x^4 - 5x^2+9\\ (x-3)^2+(y+1)^2 & = (x-3)^2 + (x^2 + 1)^2 = x^4 - 3x^2 -6x + 10\end{align}$$

The problem at hand can be rephrased as:

Given $A = (0,3)$, $B = (3,-1)$ and $P = (x,y)$ be a point on the parabola $y = x^2$. What is the minimum value of $AP + PB$?

If one make a plot of the parabola $y = x^2$, one will notice the parabola intersect with the line segment $AB$, this means the minimum value of $AP + PB$ is $$AB = \sqrt{(0-3)^2 + (3-(-1))^2} = \sqrt{3^2 + 4^2} = 5$$