Question

what are the two critical points for r=4cos( 4(theta)) and its symmetries. As well as for r=2+2sin(theta)?

Answer 1

critical points are where dr/dθ = 0.
Since r(θ) = 4 cos 4θ,
dr/dθ = -16 sin 4θ
dr/dθ = 0 at all multiples of π.
A quick look at the graph should convince you that the symmetries are
radial about the origin,
axial about the lines θ=0, π/4 and π/2

for r = 2+2sinθ,
r' = 2cosθ
so critical points are where θ=π/2, 3π/2
axial symmetry about θ=π/2