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.
- $$f ( x , y , z ) = 4 x - 3 y + 2 z , \quad x ^ { 2 } + y ^ { 2 } = 6 z$$

A) Maximum: none; minimum: $- \frac { 75 } { 4 }$ at $\left( - 6 , \frac { 9 } { 2 } , \frac { 225 } { 24 } \right)$
B) Maximum: none; minimum: $\frac { 117 } { 4 }$ at $\left( 6 , \frac { 9 } { 2 } , \frac { 225 } { 24 } \right)$
C) Maximum: none; minimum: $- \frac { 33 } { 4 }$ at $\left( 6 , \frac { 9 } { 2 } , - \frac { 225 } { 24 } \right)$
D) Maximum: none; minimum: $\frac { 225 } { 4 }$ at $\left( 6 , - \frac { 9 } { 2 } , \frac { 225 } { 24 } \right)$