  We obtain an asymptotic expansion for the tails of the random variable
$\tcal=\arg\max_{u\in\mathbb{R}}(\mathcal{A}_2(u)-u^2)$ where $\mathcal{A}_2$
is the Airy$_2$ process. Using the formula of Schehr \cite{Sch} that connects
the density function of $\tcal$ to the Hastings-McLeod solution of the second
Painlev\'e equation, we prove that as $t\rightarrow\infty$,
$\mathbb{P}(|\tcal|>t)=Ce^{-4/3\varphi(t)}t^{-145/32}(1+O(t^{-3/4}))$, where
$\varphi(t)=t^3-2t^{3/2}+3t^{3/4}$, and the constant $C$ is given explicitly.
