Suppose $f_\lambda:\mathbb{R}^3\to\mathbb{R}$ is a one-parameter family of smooth function where $\lambda\in \mathbb{R}$, which has a unique global maximum at $\mathbf{x}_0(\lambda)$. An example would be a class of Gaussian functions which has some parameter $\lambda$ in it.
Is there generic relationship between the maximum of $f_\lambda$ and the maximum of \begin{align} I(\lambda):=\int_{\mathbb{R}^3}f_{\lambda}(\mathbf{x})\,d^3\mathbf{x} \end{align} given that $f_\lambda$ is integrable and also positive, i.e. $0\leq I(\lambda)<\infty$?
I played around with Gaussian function with some parameter $\lambda$ and it seems to me that there is no relation: intuitively, I think this is analogous to the fact that the maximum of error function (which is really at $\infty$) has nothing to do with the maximum of the Gaussian (which is its antiderivative).
I wonder if there is either (1) a restricted class of smooth positive functions in which such relations exist, e.g. when the integral also has unique global maximum, or (2) an argument why such a relationship should not be expected (or ill-defined, or useless).