You could try using the parametrized equations for an ellipse rotated at an arbitrary angle:
x = h + a*cos(t)*cos(phi) - b*sin(t)*sin(phi) [1]
y = k + b*sin(t)*cos(phi) + a*cos(t)*sin(phi) [2]
…where ellipse has centre (h,k) semimajor axis a and semiminor axis b, and is rotated through angle phi.
You can then differentiate and solve for gradient = 0:
0 = dx/dt = -a*sin(t)*cos(phi) - b*cos(t)*sin(phi)
=>
tan(t) = -b*tan(phi)/a [3]
Which should give you many solutions for t (two of which you are interested in), plug that back into [1] to get your max and min x.
Repeat for [2]:
0 = dy/dt = b*cos(t)*cos(phi) - a*sin(t)*sin(phi)
=>
tan(t) = b*cot(phi)/a [4]
Lets try an example:
Consider an ellipse at (0,0) with a=2, b=1, rotated by PI/4:
[1] =>
x = 2*cos(t)*cos(PI/4) - sin(t)*sin(PI/4)
[3] =>
tan(t) = -tan(PI/4)/2 = -1/2
=>
t = -0.4636 + n*PI
We are interested in t = -0.4636 and t = -3.6052
So we get:
x = 2*cos(-0.4636)*cos(PI/4) - sin(-0.4636)*sin(PI/4) = 1.5811
and
x = 2*cos(-3.6052)*cos(PI/4) - sin(-3.6052)*sin(PI/4) = -1.5811