To understand the function and region given, let’s break down each component:

1. Function Definition: $f(x, y, z) = 1 - \sqrt{x^2 + y^2 + z^2}$ This function essentially subtracts the Euclidean distance (the square root term) from $1$. Therefore, $f(x, y, z)$ will vary depending on the distance from the origin to the point $(x, y, z)$.

2. Region $B$: $B = \{(x, y, z) \ | \ x^2 + y^2 + z^2 \leq 9, \ y \geq 0, \ z \geq 0\}$

   • $x^2 + y^2 + z^2 \leq 9$ describes a sphere with a radius of $3$.
   • The conditions $y \geq 0$ and $z \geq 0$ restrict this region to the portion of the sphere located in the first octant where both $y$ and $z$ are non-negative.

Therefore, $B$ represents the quarter-sphere (due to the first-octant restriction) with radius $3$.

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Would you like a detailed example or further breakdown of how to evaluate $f(x, y, z)$ for specific points in $B$?

Here are five related questions to explore further:

1. How does the function $f(x, y, z)$ behave as points move away from the origin?
2. What is the maximum and minimum value of $f(x, y, z)$ within the region $B$?
3. How does the restriction $y \geq 0$ and $z \geq 0$ impact the geometry of the region?
4. Could we extend $B$ by changing the inequality $x^2 + y^2 + z^2 \leq 9$ to a different radius?
5. What is the gradient of $f(x, y, z)$ and what does it imply about the function’s behavior?

Tip: When examining regions like $B$ defined by inequalities, visualize the constraints individually (like the sphere and octant restrictions) before combining them for clarity.