Find and classify the stationary points of the polynomial $$ f(x) = (x^2 + x - 11)^2 + (x^2 + x - 7)^2. $$
Using numpy.polynomial.Polynomial,
In [x]: p1 = Polynomial([-11,1,1])
In [x]: p2 = Polynomial([-7,1,1])
In [x]: p = p1**2 + p2**2
In [x]: dp = p.deriv() # first derivative
In [x]: stationary_points = dp.roots()
In [x]: ddp = dp.deriv() # second derivative
In [x]: minima = stationary_points[ddp(stationary_points) > 0]
In [x]: maxima = stationary_points[ddp(stationary_points) < 0]
In [x]: inflections = stationary_points[np.isclose(ddp(stationary_points),0)]
In [x]: print(np.array((minima, p(minima))).T)
[[-3.54138127 8. ]
[ 2.54138127 8. ]]
In [x]: print(np.array((maxima, p(maxima))).T)
[[ -0.5 , 179.125]]
In [x]: print(np.array((inflections, p(inflections))).T)
[]
That, is the function has two minima, \begin{align*} f(-3.54138127) &= 8\\ f( 2.54138127) &= 8 \end{align*} one maximum, $$ f(-0.5) = 179.125 $$ and no points of inflection / undulation.