Find the Absolute Extrema Of $$f ( x , y ) = 4 x ^ { 2 } + 2 y ^ { 2 } + 24 x - 8 \text { on the region }$$

Multiple Choice

Find the absolute extrema of $$f ( x , y ) = 4 x ^ { 2 } + 2 y ^ { 2 } + 24 x - 8 \text { on the region }$$ $$R = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 49 \right\} .$$

A) absolute minimum: $( - 3,0 , - 44 )$
absolute maximum: $( - 6 , - \sqrt { 13 } , 306 )$

B) absolute minimum: $( - 3,0 , - 44 )$
absolute maximum: $( 0,0 , - 8 )$

C) absolute minimum: $( 0,0 , - 8 )$
absolute maximum: $( 3,0,100 )$

D) absolute minimum: $( - 3,0 , - 44 )$
absolute maximum: $( 7,0,356 )$

E) absolute minimum: $( - 3,0 , - 44 )$
absolute maximum: $( - 6 , \sqrt { 13 } , 18 )$