$($a$)($i$)$ Determine whether the following vector field is conservative, $($i$.$e$.$ curl $F=0).$

$F(x,y,z)=(8x{y}^{3}z)i+(12x^2{y}^{2}z-2y{z}^3)j+(4x^2{y}^3-3y^2{z}^2)k$

$($ii$)$ Find the directional derivative of $(x,y,z)=9x^4{y}^3-8x{z}^3+5x^{2}y-6y{z}^5$ at the point $(1,1,1)$ in the direction of the vector $a=2i-j+3k$

$(15$ marks$)$

$($b$)$ Show that the following partial differential equation is exact and find its general solution

$(12x^3{y}^3-12ycos4x)dx+(9x^4{y}^2-3sin4x-42e^{-7y})dy=0$

$($c$)$ Show that the function

$f(x,y)=30x^{2}y-45x^2+4y^3-30y^2+7$

has four critical points and determine their nature.

$(40$ marks$)$