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elige la respuesta correcta entre la a,b,c,d y justifica

The image contains a problem statement written in Spanish, which is a mathematical exercise that involves analyzing the properties of a differentiable function \( f \). Let's break down the problem:

### Problem Breakdown:

1. **Given**: \( f: \mathbb{R}^2 \to \mathbb{R} \) is a differentiable function with:
   \[
   f(x,y) = C
   \]
   and the gradient of \( f \) is given by:
   \[
   \nabla f(x_0, y_0) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \neq 0
   \]

2. **Questions**:
   a) \( f(x,y) \) is constant on the vector \( \vec{v} \) that passes through the origin.
   
   b) \( \nabla f(x,y) \) is perpendicular to the vector that passes through the origin at the point \( (x_0, y_0) \).
   
   c) The maximum value of the directional derivative of \( f \) at \( (x_0, y_0) \) in the direction \( \vec{v} \) is \( 8 \).
   
   d) \( f(x_0, y_0) = 2 \) is the maximum value of the directional derivative of \( f \) at \( (x_0, y_0) \) in the direction \( \vec{v} \), and this value is \( 8 \).

### Analysis:

1. **Gradient Interpretation**: The gradient \( \nabla f(x_0, y_0) \) points in the direction of the steepest ascent, and its magnitude represents the rate of change of \( f \) in that direction. If the gradient is perpendicular to a vector \( \vec{v} \), the directional derivative in that direction is zero, meaning there's no change in \( f \) along \( \vec{v} \).

2. **Directional Derivative**: The directional derivative in the direction of a vector \( \vec{v} \) is given by:
   \[
   D_{\vec{v}}f(x_0, y_0) = \nabla f(x_0, y_0) \cdot \frac{\vec{v}}{\|\vec{v}\|}
   \]
   The maximum value of the directional derivative occurs when \( \vec{v} \) is in the direction of the gradient.

### Answer Justification:

- **Option b** is correct. The gradient \( \nabla f(x_0, y_0) \) is perpendicular to the vector \( \vec{v} \) that passes through the origin. This is consistent with the fact that the gradient is orthogonal to level curves (where \( f(x,y) \) is constant), and thus the directional derivative in the direction \( \vec{v} \) would be zero, confirming no change in \( f(x, y) \) along that direction.

Do you need any further details or clarifications on the solution? Here are some related questions to expand on this topic:

1. How does the gradient vector relate to level curves of a function?
2. What does the directional derivative represent in terms of the gradient?
3. Can the gradient ever be zero, and what does that signify?
4. How do you find the direction in which a function increases the fastest?
5. What is the geometric interpretation of a gradient vector in 3D?

**Tip**: When dealing with gradients and directional derivatives, remember that the gradient points in the direction of the maximum rate of increase of the function.

Explore the properties of differentiable functions and directional derivatives with a detailed analysis of given mathematical problems. Learn about gradients, directional derivative formulas, and their applications. Suitable for university-level mathematics.

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