Problem $6.(14$ points$)$

Consider the function

$z=f(x,y)=x^2+6x{y}^3$

$(a)$ Find $f(-5,-1).$

${?}^{\text{'}}z=f(-5,-1)=$

$(b)$ Find a function $g(x,y,z)$ whose level zero set $is$

equal $to$ the graph $ofz=f(x,y)$ and such that the

coefficient $ofzing(x,y,z)is1.$

The level set $g(x,y,z)=,=0is$ the

same $as$ the graph $ofz=f(x,y).$

$(c)$ Find the gradient $ofg.$ Write your answer $asa$

row vector $of$ the general form $($:$a,b,c$:$).$

gradg$(x,y,z)=$

$(d)$ Use gradg $to$ find a vector vec$(n)$ perpendicular $(or$ normal$)$

$to$ the graph $ofz=f(x,y)at$ the point $(-5,-1,55).$

Write your answer $as$ a row vector $of$ the general form

$($:$a,b,c$:$).$

vec$(n)=$

$(e)$ Find $an$ equation for the tangent plane $toz=f(x,y)$

$at$ the point $(-5,-1,55).$ Enter your answer $asan$ equa$-$

tion.