Find the descending directions at (2,1) for the following functions:

f( x , y ) = x + y + xy

To find the descending directions at a given point \((x_0, y_0)\) for the function \(f(x, y) = x + y + xy\), we need to compute the negative of the gradient vector at that point. The gradient vector points in the direction of steepest ascent, so its negative will indicate the steepest descent direction.

### Step 1: Compute the gradient of \(f(x, y)\)
Given the function:
\[
f(x, y) = x + y + xy
\]

The gradient \(\nabla f(x, y)\) is given by:
\[
\nabla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)
\]

Compute the partial derivatives:

For \(x\):
\[
\frac{\partial f}{\partial x} = 1 + y
\]

For \(y\):
\[
\frac{\partial f}{\partial y} = 1 + x
\]

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 \(

Learn how to find the descending directions at the point (2, 1) for the function f(x, y) = x + y + xy. This problem involves computing the gradient vector and understanding its role in determining directions of steepest descent in multivariable calculus. Suitable for college-level students and those studying multivariable calculus.

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