How many times can quadric kiss cosine at given point?

Question

Let a quadric $ax^2+2bxy+cy^2+dx+ey+f=0$ touches the plot of $y=\cos(x)$ at the point $(0,1)$ with multiplicity $n$. What is the maximum possible value of $n$? Recall that a joint point $P$ of parametrized curves $r_1(t)$ and $r_2(t)$ is said to be a tangent point of multiplicity $n$ if the first $n$ derivatives of $r_1(t)$ and $r_2(t)$ at the point $P$ are equal.

PS. That was asked as one of the problems of a student mathematical olimpiad in Russia. The author gives general directions and the answer $n=5$.

Answer 1

By symmetry the quadric we are looking for has an apex at $(0,1)$, whereby a parabola $y=1- p x^2$ obviously won't do. Therefore the equation of this quadric will be of the form $a x^2+ c y^2 +e y + f=0$ with $c\ne0$. Solving for $y$ and taking care of $y(0)=1$ we obtain an expression of the form $$y(x)=1\pm\bigl(\sqrt{q^2 + r x^2} -q\bigr)$$ with $q\geq0$ and $r$ to be determined such that $$y^{(k)}(0)=\cos^{(k)}(0)\qquad(0\leq k\leq n)\tag{1}$$ and $n$ as large as possible. Doing the computation one finds that the hyperbolic arc $$y(x)=4-\sqrt{9+3x^2}$$ and no other quadric satisfies $(1)$ with $n=5$. Since for this arc one has $$y(x)=1-{x^2\over2}+{x^4\over24}-{x^6\over144}+\ {\rm higher\ terms}$$ it follows that $(1)$ cannot be satisfied with $n=6$. Therefore $n=5$ is indeed the maximal order of kissing between a quadric and the cosine curve at $(0,1)$.

Answer 2

There are $4$ free parameters when passing a conic through a given point, so fourth-order contact for a general smooth function at a general point, but it is possible to have higher order contact at special points of a general curve, or at many points of a special curve (such as an arc of a conic).

[Edit: this corrects the value of $3$ given earlier. Increase all values in the comments by $1$ for the order of tangency.

There is a symmetry of the graph of $y=\cos x$ relative to the line $x=0$, so that the local approximating conic should have enhanced contact by order at least $1$, or order $\geq 5$ tangency. There is no reason to suppose an additional accident occurs, so order $= 5$ stated in the revised question is perfectly sensible and would say that the quality of approximation is the same as for any generic smooth even function at $x=0$. A computation is needed to check this by finding the approximating conic and verifying that there is no extra tangency more than predicted.]

[More edit: in the power series for $y = \cos x$ we can, up to $O(x^6)$ error, replace the $\frac{x^4}{24}$ by $\frac{(1-y)^2}{6}$ and this gives order $\geq 5$ tangency to a conic: $y = 1 - \frac{x^2}{2} + \frac{(1-y)^2}{6} + O(x^6)$. A computation of the $x^6$ term is needed to show that $\geq 6$ does not happen. As written earlier, there is nothing unique to $\cos x$ here, and a generic even power series would have the same order of tangency at $0$.]