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

f( x , y ) = x^2 + y^2

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

### Step 1: Compute the gradient of \(f(x, y)\)
Given the function:
\[
f(x, y) = x^2 + y^2
\]

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:
\[
\frac{\partial f}{\partial x} = 2x
\]
\[
\frac{\partial f}{\partial y} = 2y
\]

So the gradient is:
\[
\nabla f(x, y) = (2x, 2y)
\]

### Step 2: Evaluate the gradient at the point \((2,1)\)
Substitute \(x = 2\) and \(y = 1\) into the gradient:
\[
\nabla f(2, 1) = (2(2), 2(1)) = (4, 2)
\]

### Step 3: Find the descending direction
The descending direction is the negative of the gradient:
\[
-\nabla f(2, 1) = (-4, -2)
\]

### Conclusion:
The descending direction at the point \((2, 1)\) for the function \(f(x, y) = x^2 + y^2\) is given by the vector \((-4, -2)\).

This vector indicates the direction in which the function decreases most rapidly at the point \((2,1)\).

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Would you like to know more details or have any questions? Here are 8 follow-up questions related to gradients and directional derivatives:

1. What is the significance of the gradient vector in a multivariable function?
2. How do you compute the directional derivative of a function in a given direction?
3. What does the magnitude of the gradient vector tell us about the function's behavior?
4. How can you find the direction of steepest ascent for a given function at a point?
5. What role does the Hessian matrix play in analyzing the behavior of multivariable functions?
6. How do you determine whether a critical point is a maximum, minimum, or saddle point?
7. What is the connection between the gradient and level curves of a function?
8. How can you visualize the gradient vector field for a given function?

**Tip:** The gradient vector not only gives the direction of the steepest ascent but its magnitude also indicates the rate of increase in that direction.

Learn how to find the descending directions at the point (2, 1) for the function f(x, y) = x^2 + y^2. Understand the concept of gradients, directional derivatives, and their application in multivariable calculus. Suitable for advanced college-level mathematics.

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