https://file.aiquickdraw.com/mathbot/file/1726321226635ydpuuzhc.jpg

The problem in the image is:

Given \( u = ax^2 + by^2 + cz^2 \), and \( u_{xx} + u_{yy} + u_{zz} = 0 \), prove that \( a + b + c = 0 \).

### Solution:

We need to find the second partial derivatives of \( u \) with respect to \( x \), \( y \), and \( z \).

1. **Expression for \( u \):**

   \[
   u = ax^2 + by^2 + cz^2
   \]

2. **Second partial derivatives:**

   - For \( u_x \) and \( u_{xx} \):
     \[
     u_x = \frac{\partial u}{\partial x} = 2ax
     \]
     \[
     u_{xx} = \frac{\partial^2 u}{\partial x^2} = 2a
     \]

   - For \( u_y \) and \( u_{yy} \):
     \[
     u_y = \frac{\partial u}{\partial y} = 2by
     \]
     \[
     u_{yy} = \frac{\partial^2 u}{\partial y^2} = 2b
     \]

   - For \( u_z \) and \( u_{zz} \):
     \[
     u_z = \frac{\partial u}{\partial z} = 2cz
     \]
     \[
     u_{zz} = \frac{\partial^2 u}{\partial z^2} = 2c
     \]

3. **Summing the second derivatives:**
   According to the condition \( u_{xx} + u_{yy} + u_{zz} = 0 \):
   \[
   2a + 2b + 2c = 0
   \]

4. **Simplifying:**
   \[
   a + b + c = 0
   \]

Thus, we have proved that \( a + b + c = 0 \).

Would you like more details on this process or any related concepts? Here are some follow-up questions that might deepen your understanding:

1. What are the first-order partial derivatives of \( u \) with respect to \( x \), \( y \), and \( z \)?
2. How do partial differential equations (PDEs) relate to this problem?
3. Can you find another function \( u \) that satisfies the condition \( u_{xx} + u_{yy} + u_{zz} = 0 \)?
4. What does this result mean geometrically in terms of the surface described by \( u \)?
5. How can this technique be applied to other quadratic forms?

**Tip:** Understanding the role of partial derivatives is crucial for solving PDEs and other problems in multivariable calculus.

If u = ax^2 + by^2 + cz^2 and u_xx + u_yy + u_zz = 0, then prove that a + b + c = 0.

Learn how to prove that a + b + c = 0 given u = ax^2 + by^2 + cz^2 and u_xx + u_yy + u_zz = 0. This step-by-step solution covers partial derivatives and quadratic forms, suitable for advanced high school or undergraduate students.

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