I have perhaps a very simple question where I lack some inutition at the moment: Is the expression $$\sup_{\alpha < 0, \lambda \in \mathbb N}\int_{-\infty}^{\alpha} e^{-\lambda t^4} \ dt \int_{\alpha}^0 e^{\lambda t^4} \ dt $$

finite or not?-My main concern is $\alpha$ close to zero since $t^4$ is not strongly convex.

I have perhaps a very simple question where I lack some inutition at the moment: Is the expression $$\sup_{\alpha < 0, \lambda \in \mathbb N}\int_{-\infty}^{\alpha} e^{-\lambda t^4} \ dt \int_{\alpha}^0 e^{\lambda t^4} \ dt $$

finite or not?-My main concern is $\alpha$ close to zero since $t^4$ is not strongly convex (this is something that would've bailed me out I think, so for Gaussians I think I could show this boundedness).

I have perhaps a very simple question where I lack some inutition at the moment: Is the expression $$\sup_{\alpha < 0, \lambda \in \mathbb N}\int_{-\infty}^{\alpha} e^{-\lambda t^4} \ dt \int_{\alpha}^0 e^{\lambda t^4} \ dt $$

finite or not?-My main concern is $\alpha$ close to zero...

I have perhaps a very simple question where I lack some inutition at the moment: Is the expression $$\sup_{\alpha < 0, \lambda \in \mathbb N}\int_{-\infty}^{\alpha} e^{-\lambda t^4} \ dt \int_{\alpha}^0 e^{\lambda t^4} \ dt $$

finite or not?-My main concern is $\alpha$ close to zero since $t^4$ is not strongly convex.