Which of the following functions corresponds to the level curve in the following graph?

Antwoord 1 correct

Correct

Antwoord 2 optie

$z(x,y)=2x + 4y$.

Antwoord 2 correct

Fout

Antwoord 3 optie

$z(x,y)=\min\{4x,2y\}$.

Antwoord 3 correct

Fout

Antwoord 4 optie

$z(x,y) = \min\{x,2y\}$.

Antwoord 4 correct

Fout

Antwoord 1 optie

$z(x,y)=\min\{2x,4y\}$.

Antwoord 1 feedback

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.

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.

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