Find the minimum of function

Question

Find the minimum of $f(x)=\sqrt{(1-x^2)^2+(2-x)^2}+\sqrt{x^4-3x^2+4}$.
I haven't learned derivative so I tried to solve it with geometry.But in the end I failed.
Maybe it can be seemed as the sum of the lengths of two line segments?

Answer 1

$$f(x)=\sqrt{(x-2)^2+(x^2-1)^2}+\sqrt{(x)^2+(x^2-2)^2}$$

So $f(x)$ is the sum of the distances from a point on the parabola $y=x^2$ to points $(2,1)$ and $(0,2)$.

$(0,2)$ is inside the parabola and $(2,1)$ is outside, so the point $(x,x^2)$ that makes $f(x)$ minimum would lie on the line segment that connects the two points.

Therefore, the minimum of $f(x)$ is $\sqrt{5}$.