Answer
$$K = \frac{1}{{{{\left( {17} \right)}^{3/2}}}},{\text{ }}r = 17\sqrt {17} $$
Work Step by Step
$$\eqalign{
& y = \frac{1}{2}{x^2} + 2,{\text{ }}x = 4 \cr
& {\text{Calculate the curvature, use }}K = \frac{{\left| {y''\left( x \right)} \right|}}{{{{\left( {1 + {{\left[ {y'\left( x \right)} \right]}^2}} \right)}^{3/2}}}} \cr
& {\text{Find }}y'\left( x \right){\text{ and }}y''\left( x \right){\text{ }} \cr
& y'\left( x \right) = \frac{d}{{dx}}\left[ {\frac{1}{2}{x^2} + 2} \right] \cr
& y'\left( x \right) = x \cr
& {\text{Evaluate at }}x = 4 \cr
& y'\left( 4 \right) = 4 \cr
& y''\left( x \right) = \frac{d}{{dx}}\left[ x \right] \cr
& y''\left( x \right) = 1 \cr
& {\text{Evaluate at }}x = 4 \cr
& y''\left( 4 \right) = 1 \cr
& \underbrace {K = \frac{{\left| {y''\left( x \right)} \right|}}{{{{\left( {1 + {{\left[ {y'\left( x \right)} \right]}^2}} \right)}^{3/2}}}}}_ \Downarrow \cr
& {\text{ at }}x = 4 \cr
& K = \frac{{\left| 1 \right|}}{{{{\left( {1 + {{\left[ 4 \right]}^2}} \right)}^{3/2}}}} \cr
& K = \frac{1}{{{{\left( {17} \right)}^{3/2}}}} \cr
& {\text{The radius of curvature is }}r = \frac{1}{K} \cr
& r = {17^{3/2}} \cr
& r = 17\sqrt {17} \cr} $$
