$(25$ points total$)$ Consider the following vector field:

vec$(F)(x,y)=(-y^2)vec(i)+(2x)vec(j)$

$($a$)(6$ points$)$ Find the maximum magnitude of a vector in the vector field vec$(F)$ such that the vector originates at a point on the curve $x^2+3y^2=5.($Hint: This is an optimization problem, not really a vector field problem.$)($Hint #$2$: It is sufficient to maximize the square of the magnitude.$)$

$($b$)(2$ points$)$ Find the vectors of vec$(F)$ corresponding to the points $(1,0$