To find the descending directions at the point $(2,1)$ for the function $f(x, y) = x + y + xy$, we need to follow these steps:

Step 1: Compute the Gradient

The gradient of the function $f(x, y)$ is a vector that points in the direction of the steepest ascent. It is given by:

$\nabla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)$

Let's compute the partial derivatives.

1. Partial derivative with respect to $x$: $\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left(x + y + xy\right) = 1 + y$

2. Partial derivative with respect to $y$: $\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \left(x + y + xy\right) = 1 + x$

So, the gradient is:

$\nabla f(x, y) = (1 + y, 1 + x)$

Step 2: Evaluate the Gradient at the Point $(2,1)$

Substitute $x = 2$ and $y = 1$ into the gradient:

$\nabla f(2, 1) = (1 + 1, 1 + 2) = (2, 3)$

Step 3: Determine the Descending Direction

The descending direction is opposite to the gradient vector, so it will be:

$-\nabla f(2, 1) = (-2, -3)$

Conclusion

The descending direction at the point $(2, 1)$ for the function $f(x, y) = x + y + xy$ is given by the vector $(-2, -3)$.

Would you like further details or have any other questions?

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Here are some related questions to expand your understanding:

1. What is the significance of the gradient in multivariable calculus?
2. How do you find the direction of steepest ascent for a function?
3. Can you explain why the descending direction is opposite to the gradient?
4. How would the calculation change if the function included more variables?
5. What are the real-world applications of finding descending directions in functions?
6. How do you normalize a gradient vector, and why might that be useful?
7. What would happen if you evaluated the gradient at different points?
8. How does the shape of the function's graph relate to the gradient?

Tip: Remember that the gradient not only points in the direction of the steepest ascent but also its magnitude indicates the rate of change in that direction.