Find the Extreme Values of the Function Subject to the Given

Multiple Choice

Find the extreme values of the function subject to the given constraint.
- $$f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } , x + 2 y + 3 z = 6$$

A) Maximum: none; minimum: $\frac { 72 } { 7 }$ at $\left( \frac { 6 } { 7 } , \frac { 12 } { 7 } , \frac { 18 } { 7 } \right)$
B) Maximum: none; minimum: $\frac { 2 } { 7 }$ at $\left( \frac { 1 } { 7 } , \frac { 2 } { 7 } , \frac { 3 } { 7 } \right)$
C) Maximum: none; minimum: 2 at $\left( \frac { 1 } { 7 } , \frac { 4 } { 7 } , \frac { 9 } { 7 } \right)$
D) Maximum: none; minimum: $\frac { 18 } { 7 }$ at $\left( \frac { 3 } { 7 } , \frac { 6 } { 7 } , \frac { 9 } { 7 } \right)$