Answer
$$\left( {\frac{1}{2}, - 1, - \frac{{31}}{4}} \right)$$
Work Step by Step
$$\eqalign{
& z = 3{x^2} + 2{y^2} - 3x + 4y - 5 \cr
& 3{x^2} + 2{y^2} - 3x + 4y - 5 - z = 0 \cr
& {\text{Consider}} \cr
& F\left( {x,y,z} \right) = 3{x^2} + 2{y^2} - 3x + 4y - 5 - z \cr
& {\text{Calculating the partial derivatives}} \cr
& {F_x}\left( {x,y,z} \right) = 6x - 3 \cr
& {F_y}\left( {x,y,z} \right) = 4y + 4 \cr
& {F_z}\left( {x,y,z} \right) = - 1 \cr
& {\text{Find the gradient }}\nabla F\left( {x,y,z} \right) \cr
& \nabla F\left( {x,y,z} \right) = \left( {6x - 3} \right){\bf{i}} + \left( {4y + 4} \right){\bf{j}} - {\bf{k}} \cr
& {\text{Find the point}}\left( {\text{s}} \right){\text{ on the surface at which the tangent plane }} \cr
& {\text{is horizontal}}. \cr
& 6x - 3 = 0 \to x = 1/2 \cr
& 4y + 4 = 0 \to y = - 1 \cr
& z = 3{x^2} + 2{y^2} - 3x + 4y - 5 \cr
& z = 3{\left( {1/2} \right)^2} + 2{\left( { - 1} \right)^2} - 3\left( {1/2} \right) + 4\left( { - 1} \right) - 5 \cr
& z = - 31/4 \cr
& {\text{The point is}} \cr
& \left( {\frac{1}{2}, - 1, - \frac{{31}}{4}} \right) \cr} $$
