1. The problem is to understand the function $f(x,y,z) = x + y + z$ defined over a rectangular box in 3D space with corners at $(0,0,0)$ and $(1,2,3)$. 2. This function simply adds the coordinates $x$, $y$, and $z$ of any point inside or on the box. 3. Important to note: since $x$, $y$, and $z$ are all non-negative within the box, $f(x,y,z)$ will range from $0$ at the origin to $1 + 2 + 3 = 6$ at the opposite corner. 4. To evaluate $f$ at any point, just substitute the coordinates into the formula. For example, at $(1,0,0)$, $f = 1 + 0 + 0 = 1$. 5. The function increases linearly along any edge or inside the box because it is a sum of the coordinates. 6. The contour line $C$ on one face represents points where $f$ is constant, i.e., $x + y + z = ext{constant}$. 7. The orange segment with arrows inside the box likely indicates a direction of increasing $f$. Final answer: The function $f(x,y,z) = x + y + z$ increases linearly from $0$ at $(0,0,0)$ to $6$ at $(1,2,3)$ inside the box.