I want to know what I can say about the following problem:
$$\underset{y}{\text{maximize}} \; K \int_{y}^{\infty} f^{2}(x) dx - \int_{y}^{\infty} g^{2}(x) dx $$
where $K \in \mathbb{R}_{+}$, $y \geq 0$, and $f,g \in \mathbb{R}$, and $f^{2}(x)$ $g^{2}(x)$ are decreasing functions.
It is clear that $\int_{y}^{\infty} f^{2}(x) dx$ and $\int_{y}^{\infty} g^{2}(x) dx$ are decreasing functions of $y$.
Questions:
- Can I solve this problem by simply finding:
$$ \frac{d}{d y} \left( K \int_{y}^{\infty} f^{2}(x) dx - \int_{y}^{\infty} g^{2}(x) dx \right) = 0 $$
** Or do I need other restrictions on these functions?
If I do this, knowing the decreasing nature of these functions (ie $ \underset{x \rightarrow \infty}{\text{lim}} \; f^{2}(x) = g^{2}(x) = 0)$ I get:
$$ g^{2}(y) - K f^{2}(y)= 0 $$
Then, I could find the result numerically -- assuming I know the functional forms of $f,g$.
Is this approach reasonable? Will it solve the original problem?
Thanks!