Inequality with Arc Length

Question

Define $L:=\int_{0}^{1}\sqrt{(x'(t))^2+(y'(t))^2} dt$.

Show $L\geq|x(1)-x(0)|$.

I don't know where to start.

Answer 1

Hint: $$\int_{0}^{1}\sqrt{(x'(t))^2+(y'(t))^2} dt \ge \int_{0}^{1}\sqrt{(x'(t))^2} dt = \int_{0}^{1} \lvert x'(t) \rvert dt $$

Answer 2

What happens if you replace the $y'(t)$ term by $0?$