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Let $$\begin{aligned} f(x,y) & = 2 x^6 - x^4 y^2 - x^2 y^4 + 2 y^6 \\ & = x^6 + y^6 + (x^2 - y^2)^2 (x^2 + y^2) . \end{aligned}$$ Then $f$ is a strictly positive (except at the origin, of course) homogeneous polynomial of degree $6$, and hence $d^j f = 0$ for $j < 6$ and $d^6 f > 0$. On the other hand, $$\partial_{xx} f(0,y) = -2 y^4 < 0$$ whenever $y \ne 0$, and so $f$ is not convex near $0$.