Hello everyone, while I was at a leadership conference my mind wandered and I found myself pondering the optimization of functions of the form

$\displaystyle f(\theta)=\int_a^{b}f(x,\theta)~dx\quad\text{where }a\ne{b}$

So I basically treated it as a function of one variable, and proceeding as such I arrived at a question.

$\displaystyle f'(\theta)=\frac{d}{d\theta}\int_a^{b}f(x,\theta)~ dx$

Now by Leibniz's rule the above equation is equivalent to

$\displaystyle f'(\theta)=\int_a^{b}\frac{\partial{f(x,\theta)}}{ \partial\theta}~dx$

Now I next looked for critical points.

If we let $\displaystyle F(\theta)=\int_a^{b}\frac{\partial{f(x,\theta)}}{\ partial\theta}~dx$

Then the critical points occur at four points, namely :

When $\displaystyle F(\theta)=0$

Where $\displaystyle F(\theta)$ is undefined

Less obviously, where $\displaystyle \frac{\partial{f(x,\theta)}}{\partial\theta}=0$. This obviously arises from the fact that $\displaystyle \int_a^{b}0~dx=0$

And also where $\displaystyle \frac{\partial{f(x,\theta)}}{d\theta}$ is undefined.

Now here is where I ran into an anomaly, I found multiple cases where $\displaystyle c\in\mathbb{R}$, satisfies two conatradicting critical point criteria, namely.

$\displaystyle F(c)$ is undefined, but $\displaystyle \frac{\partial{f(x,c)}}{\partial\theta}=0$.

So this implies that this is a multi-valued function, so what does one do from here? Does one just sweep this doubly critical point under the carpet and continue? I hope not.

Any guidance would be appreciated.

Alex