1. Given that $f(x, y) = x^2y + 3x^2$, find its derivative with respect to $x$, i.e, find $\frac{\partial f}{\partial x}$.
Note: Please use * to indicate the product in the answer. So, if we would write the entire function $f$ as an answer, it would be x^2 * y + 3 * x^2.
Answer: $2xy + 6x$ (2 * x * y + 6 * x)
2. Given that $f(x, y) = xy^2 + 2x + 3y$ its gradient, i.e., $\nabla f(x, y)$ is:
$$\begin{bmatrix} 2xy + 3 \cr y^2 + 2 \end{bmatrix}$$
$$\begin{bmatrix} 2xy \cr 2x + 3 \end{bmatrix}$$
$$\begin{bmatrix} y^2 + 2 \cr 2xy + 3 \end{bmatrix}$$
$$\begin{bmatrix} 2y \cr 0 \end{bmatrix}$$
3. Let $f(x, y) = x^2 + 2y^2 + 8y$. The minimum value of $f$ is:
Hint: The question asks for the minimum value that the function can output, and not the point $(x, y)$ that gives it.
Answer: -8
4. The gradient of $f(x, y, z) = x^2 + 2xyz + z^2$ is:
$$\begin{bmatrix} 2x + 2yz \cr 2xz \cr 2xy + 2z \end{bmatrix}$$
$$\begin{bmatrix} 2x + 2xz \cr 2yz \cr 2xy + z \end{bmatrix}$$
$$\begin{bmatrix} 2x + 2yz \cr 2xy \cr 2xy + z \end{bmatrix}$$
$$\begin{bmatrix} 2yz + 2xz \cr 2z \cr 2x \end{bmatrix}$$