Answer
$$\left( {0,0} \right)$$
Work Step by Step
$$\eqalign{
& f\left( {x,y} \right) = 1 + {x^2} + {y^2} \cr
& {\text{Calculate the partial derivatives }}{f_x}\left( {x,y} \right){\text{ and }}{f_y}\left( {x,y} \right) \cr
& {f_x}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {1 + {x^2} + {y^2}} \right] = 2x \cr
& {f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {1 + {x^2} + {y^2}} \right] = 2y \cr
& {\text{Set }}{f_x}\left( {x,y} \right) = 0{\text{ and }}{f_y}\left( {x,y} \right) = 0 \cr
& {f_x}\left( {x,y} \right) = 0 \cr
& 2x = 0 \cr
& x = 0 \cr
& and \cr
& {f_y}\left( {x,y} \right) = 0 \cr
& 2y = 0 \cr
& y = 0 \cr
& {\text{The critical point occurs at}} \cr
& \left( {0,0} \right) \cr} $$
