I have the following question:
Let $c>0$ and consider the following function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ given by:
$$f(x,y)=\begin{cases} c y e^{-x} & \text{0 < x < $\infty$ and 0 < y < $e^{-x}$} \\ 0 & \text{else} \\ \end{cases} $$
- "Sketch"/show the area where $f$ is strictly positive.
By the intervals i notice that $y \in (0,1)$ because it's dependent of $x$.
I've plotted the graph, as shown here, and there is a curved area where $f(x,y)>0$, but i have no idea how i formally show or specify the area as an interval or "sketch" it.
I'm pretty stuck on this one, so i hope there is someone who can get me a little push.
First, we know that $0$ is not strictly positive. So, if we don't have $0 < x < \infty$ and $0 < y < e^{-x},$ then $f(x, y)$ is equal to $0$, hence is not strictly positive.
Now, suppose we do have $0 < x < \infty$ and $0 < y < e^{-x}.$ Note that $e^{-x}$ is always strictly positive, $y$ is always strictly positive in this range ($0 < y < e^{-x}$), and that $c$ is strictly positive by assumption. So, $cye^{-x}$ is always strictly positive when $x$ and $y$ are in this range.
So, when we have $0 < x < \infty$ and $0 < y < e^{-x},$ $f(x, y)$ is equal to $cye^{-x},$ and is thus strictly positive.
Therefore, the area where $f$ is strictly positive is just the area in the plane described by $0 < x < \infty$ and $0 < y < e^{-x}.$ So you just need to sketch this.