Maximum value of $f(x,y)=e^{-y} - \ln x$ on $(x,y) \in (0,1) \times (0,1).$

Question

Let us consider a function $~f(x,y)$ defined by $$f(x,y) = e^{-y} - \ln x,~~ (x,y) \in [0,1] \times [0,1].$$ Now calculate the minimum and maximum of $f(x,y)$ on $[0,1] \times [0,1]$.

My Attempt: For minimum, $~\lim_{y \to 1} e^{-y} = \frac{1}{e}$ and $~\lim_{x \to 1} -\ln x = 0$ and this follows that minimum value of $f(x,y)$ is $\frac{1}{e}.$ Now I am not able to find maximum but the answer is given $2$. Please help me to find this.

Answer 1

The function does not have (absolute) maximum because it is unbounded above (note that $f(x,y)\rightarrow +\infty$ as $x\rightarrow 0$).