Find the Maximum Rate of Change Of $f ( x , y ) = x y ^ { 2 } + \sqrt { y }$

Multiple Choice

Find the maximum rate of change of $f ( x , y ) = x y ^ { 2 } + \sqrt { y }$ at the point $( 4,1 )$ (2,1) . In what direction does it occur?

A) $\frac { \sqrt { 29 } } { 2 } , \left\langle1 , \frac { 5 } { 2 } \right\rangle$
B) $\frac { 85 } { 2 } , \left\langle2 , \frac { 7 } { 2 } \right\rangle$
C) $\frac { 29 } { 2 } , \left\langle 1 , \frac { 9 } { 2 } \right\rangle$
D) $\frac { \sqrt { 293 } } { 2 } , \left\langle1 , \frac { 17 } { 2 } \right\rangle$
E) none of these