Consider the function

$F(x_1,x_2,x_3)=3x_1^2+6x_1{x}_2-2x_2^2+3x_3^3$

$($a$)$ Determine the gradient vector and Hessian matrix.

$($b$)$ Derive the first three terms of the Taylor series expansion about $(0,1,1).$

$($c$)$ What linear system should be solved for an approximation to the minimum point for $F?$