Question $3(25$ marks$)$

$($a$)$ Determine the directional derivative of $f(x,y)=x^2+xy$ in the direction vec$()+2$vec$()$ at the point $(1,1).$

$[10$ Marks$]$

$($b$)$ A company is designing a hill$-$shaped storage unit with a temperature distribution modeled by $T(x,y)=5x^2+3xy+y^2,$ where $T(x,y)$ is the temperature $($in $C)$ at a point $(x,y)$ on the surface.

At the point $(2,1)$

i$.$ Find the direction in which the temperature increases most rapidly. $[10$ Marks$]$

ii$.$ Calculate the maximum rate of change of temperature at this point. $[5$ Marks$]$