Plan curve with zero area has at least two points of zero curvature

Question

Let $\alpha=(x,y)$ be a smooth closed plan curve defined on $[a,b]\subset \mathbb{R}$. We can define the oriented area of $\alpha$ by $A=\int_{a}^{b} x(s)y'(s)ds$. So, if A=0 then there exists $t_1 \neq t_2 \in [a,b] $ such that $k(t_1)=k(t_s)=0.$ Is it true?