Math Solutions

(See Solution) One type of multivariable function we'll see a lot in this class has the form f(x, y)=min 3 x, y In other words, it returns the lesser value

Question: One type of multivariable function we'll see a lot in this class has the form

\[f(x, y)=\min \{3 x, y\}\]

In other words, it returns the lesser value of $3 x$ and \(y\). Therefore, another way we might write it is

\[f(x, y)= \begin{cases}3 x & \text { if } 3 x \leq y \\ y & \text { if } 3 x \geq y\end{cases}\]

For example:

\[\begin{array}{r} f(1,9)=\min \{3 \times 1,9\}=3 \\ f(2,9)=\min \{3 \times 2,9\}=6 \\ f(3,9)=\min \{3 \times 3,9\}=9 \\ f(4,9)=\min \{3 \times 4,9\}=9 \\ f(3,12)=\min \{3 \times 3,12\}=9 \end{array}\]
  1. Plot the level sets for \(f(x, y)=15\) and \(f(x, y)=30\).
  2. Find the partial derivatives. Hint: you probably want to write them as:
  3. Recall that the implicit function defined by \(f(x, y)=z\) has a slope of
\[\left.\frac{d y}{d x}\right|_{f(x, y)=z}=-\frac{\partial f / \partial x}{\partial f / \partial y}\]

Find the slope of the level set of this function at a generic point \((x, y)\) by plugging your answers from (b) into this formula. How does your answer relate visually to your answer to (a)?

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