The function given in part (v) is:

$f(x, y) = \sin(x) + \sin(y) + \sin(x + y)$

To solve this, let’s analyze what we can do with the function.

1. Simplifying $f(x, y)$

There are no immediate algebraic simplifications that can be applied. However, depending on the context (finding critical points, gradients, etc.), the following steps can be applied.

2. Partial Derivatives of $f(x, y)$

If you want to find critical points, we can take the partial derivatives of $f(x, y)$ with respect to $x$ and $y$.

• Partial derivative with respect to $x$:

$\frac{\partial}{\partial x} f(x, y) = \cos(x) + \cos(x + y)$

• Partial derivative with respect to $y$:

$\frac{\partial}{\partial y} f(x, y) = \cos(y) + \cos(x + y)$

3. Setting Partial Derivatives to Zero (for critical points)

To find critical points, set each partial derivative equal to zero:

$\cos(x) + \cos(x + y) = 0$ $\cos(y) + \cos(x + y) = 0$

Solving this system of equations would give the critical points, depending on the specific values of $x$ and $y$.

Let me know if you'd like to proceed with solving the system or go in another direction, such as evaluating the function at specific points.

Do you have any questions?

Here are 5 related questions:

1. Would you like to explore how to solve this system of trigonometric equations?
2. Do you need to find local minima or maxima of the function $f(x, y)$?
3. Are you interested in understanding how to interpret the critical points geometrically?
4. Would you like me to help graph the function $f(x, y)$?
5. Do you need further simplifications for specific values of $x$ and $y$?

Tip: When solving trigonometric equations, using unit circle properties can help simplify complex expressions!