Exercise 5

Correct: The shape of the level curve indicates that we are dealing with a minimum function. Hence, we look for a function $z(x,y) = \min\{ax,by\}$ of which $a$ and $b$ have to be determined.

You can see that $ax = by$ at $(4,2)$, or
$$a\cdot 4 = b\cdot2.$$
Moreover, we know that the value of $z(x,y)$ at that point is equal to 8, hence
$$ \min\{a\cdot 4, b\cdot 2\} = 8 \iff 4a = 8 \text{~and~} 2b = 8 \iff a = 2 \text{~and~} b=4.$$
Hence, the function that results in this level curve is
$$z(x,y) = \min\{2x,4y\}.$$
You can use the level curve with value $k=12$ to check your answer. The level curve with value 12 corresponding to the function $\min\{2x,4y\}$ is
$$
\left\{\begin{array}{ll}
2x = 12 & \text{if~}4y\geq12\\
4y = 12 & \text{if~}2x\geq 12
\end{array}\right.
\iff
\left\{\begin{array}{ll}
x = 6 & \text{if~}y\geq3\\
y = 3 & \text{if~}x\geq6
\end{array}\right.
$$
This is exactly the level curve with value 12 as drawn in the figure.

Go on.

Wrong: What type of function has such level curves?

See Example 1, Example 2 and in particular Example 3.

Wrong: We are indeed dealing with a minimum function $\min\{ax,by\}$, but the value of $a$ ($b$) is not given by the point of intersection with the $x$-axis ($y$-axis).

See Example 1, Example 2 and in particular Example 3.

Wrong: Pay attention to the values of the level curves.

See Example 1, Example 2 and in particular Example 3.