How to calculate uncertanty of extremal point of fitted quadratic polynom? I am doing some fitting in python, that is:
fun = np.array([1.48706e-05, 1.63946e-05, 1.61683e-05, 1.33775e-05])
var = np.array([15.368, 15.504, 15.64 , 15.776])

x = np.linspace (var[0], var[-1], 100)

(a, b, c), cov = np.polyfit(var, fun, 2, cov = True)
a,b,c
(-5.832071799333984e-05, 0.001812880441184612, -0.014071596430063667)

s_a, s_b, s_c = np.sqrt(np.diag(cov))
s_a, s_b, s_c
(4.9216223166178695e-06, 0.00015328017457825164, 0.0011933529668798418)

plt.scatter (var, fun)
plt.plot (x, a*x**2+b*x+c)
plt.show ()

Ok, so I get expected fit:

Then, I need to calculate the position of the maximum of the poly and its uncertainty. First step is easy: $x_{max}(a, b) = -\frac{\overline{b}}{2\overline{a}}\approx 15.54233$.
My attempt for uncertainty is standard error propagation:
 $\approx 1.85666$
Problem is that I expected a much smaller value for the uncertainty of $x_{max}$, something like ~0.05. Is my expectation unreasonable or did I made an error in calculation?

Of course, I made an error - polynomial coefficients are not independent variables but correlated parameters and "standard" error propagation is missing a term.

When cross-correlated (third) term is calculated, uncertainty comes out at expected order.
