Answer
$$\eqalign{
& \left( {\bf{a}} \right)L\left( {x,y} \right) = 8x - 64y + 144,\,\,\, \cr
& \left( {\bf{b}} \right)L\left( { - 1.05,3.95} \right) = - 117.2 \cr} $$
Work Step by Step
$$\eqalign{
& f\left( {x,y} \right) = 12 - 4{x^2} - 8{y^2};\,\,\,\,\left( { - 1,4} \right)\,\,\,{\text{ and }}\,\,\,\left( { - 1.05,3.95} \right) \cr
& f\left( { - 1,4} \right) = 12 - 4{\left( { - 1} \right)^2} - 8{\left( 4 \right)^2} \cr
& f\left( { - 1,4} \right) = - 120 \cr
& {\text{The partial derivatives are}} \cr
& {f_x}\left( {x,y} \right) = - 8x \cr
& {f_y}\left( {x,y} \right) = - 16y \cr
& {\text{Evaluate the partial derivatives at }}\left( { - 1,4} \right) \cr
& {f_x}\left( { - 1,4} \right) = 8 \cr
& {f_y}\left( { - 1,4} \right) = - 64 \cr
& \cr
& \left( a \right) \cr
& {\text{The linear approximation to the function at }}\,\left( { - 1,4, - 120} \right){\text{ is}} \cr
& L\left( {x,y} \right) = {f_x}\left( {a,b} \right)\left( {x - a} \right) + {f_y}\left( {a,b} \right)\left( {y - b} \right) + f\left( {a,b} \right) \cr
& L\left( {x,y} \right) = 8\left( {x + 1} \right) - 64\left( {y - 4} \right) - 120 \cr
& L\left( {x,y} \right) = 8x + 8 - 64y + 256 - 120 \cr
& L\left( {x,y} \right) = 8x - 64y + 144 \cr
& \cr
& \left( b \right){\text{Estimate }}\,\left( { - 1.05,3.95} \right) \cr
& L\left( { - 1.05,3.95} \right) = 8\left( { - 1.05} \right) - 64\left( {3.95} \right) + 144 \cr
& L\left( { - 1.05,3.95} \right) = - 117.2 \cr} $$
