Hi why there are 3 missing solutions in S4 ?
forget()
t = var('t')
f_x = function('f_x')(t)
f_y = function('f_y')(t)
assume(t, 'real')
assume(t > 0)
assume(t < 2*pi)
f_x=cos(t)
f_y=1/2 *sin(2*t)
S4=solve(derivative(f_y,t)/derivative(f_x,t)==0,t)
show(S4)
fP=parametric_plot( (f_x, f_y), (t,0,2*pi))
for s in S4 :
p=point2d((f_x(t=s.rhs()),f_y(t=s.rhs())),size=30, rgbcolor=hue(0.75), marker='d', markeredgecolor='red')
#p2=point2d((f_x(t=s.rhs()+pi/2),f_y(t=s.rhs()+pi/2)),size=30, rgbcolor=hue(0.75), marker='d', markeredgecolor='red')
#p3=point2d((f_x(t=s.rhs()+pi),f_y(t=s.rhs()+pi)),size=30, rgbcolor=hue(0.75), marker='d', markeredgecolor='red')
#p4=point2d((f_x(t=s.rhs()+3*pi/2),f_y(t=s.rhs()+3*pi/2)),size=30, rgbcolor=hue(0.75), marker='d', markeredgecolor='red')
#show(fP+p+p2+p3+p4)
show(fP+p)
