The Parent's Ultimatum

A parent threatens a screaming child: 'If you don't stop, we turn around and go home.' Model it as a sequential game. The child moves first: Quiet Down or Keep Screaming. Then the parent moves: Carry On to the beach or Drive Home. Payoffs (parent, child):

• Quiet Down, Carry On: (10, 5)
• Keep Screaming, Carry On: (4, 8)
• Keep Screaming, Drive Home: (x, 0)

(After Quiet Down the parent always Carries On.) Here $x$ is the parent's payoff from turning around, currently $x=1$. Both players are rational and payoff-maximising, and the parent breaks ties by Carrying On.

What is the smallest integer value of $x$ for which the threat becomes credible, so that in the subgame-perfect equilibrium the child Quiets Down?