$($d$)(4$ points$)$ Assume the given points, $(0,4),(0,0),(4\sqrt[\phantom{2}]{3},0)$ and $(0,-4)$ are known to be critical points of the function $f(x,y)=x^{2}y+y^3-48y.$ Circle the correct choice for each part. Show all work to support your answer.

i$.f(x,y)$ has a minimum $/$ a maximum $/$ a saddle $/$ inconclusive at $(0,4).$

ii$.f(x,y)$ has a minimum $/$ a maximum $/$ a saddle $/$ inconclusive at $(0,0).$

iii. $f(x,y)$ has a minimum $/$ a maximum $/$ a saddle $/$ inconclusive at $(4\sqrt[\phantom{2}]{3},0).$

iv$.f(x,y)$ has a minimum a maximum $/$ a saddle $/$ inconclusive at $(0,-4).$

$($e$)(4$ points$)$ Find $($i$)$ the maximum rate of decrease $f(x,y,z)=y+\frac{z}{x}$ at the point $(1,-1,0)$ and $($ii$)$ the direction in which it occurs.

gradf$=($:$-\frac{2}{x^2},1,\frac{1}{x}$:$)$

gradf$(1,-1,0)=($:$-\frac{0}{1^2},1,\frac{1}{1}$:$)$

gradf$=($:$0,1,1$:$)|||$

$\sqrt[\phantom{2}]{0^2+1^2+1^2}=\frac{\sqrt[\phantom{2}]{2}}{\text{u}\text{a}\text{r}\text{r}}in$ dinection $x-2$

This $is$ Direction must $be$

a vector

$(-2)$

max D$$f$.$Please fix my work for part a and b