I am trying to show that $f(x,y) = \sin(x^2+y^2)+\cos(x+y)$ has a local minimum at the origin, via the following Taylor expression (centered at the origin): $f(x,y) = 1 +\frac{(x-y)^2}{2} + R_2(x,y)$, with $\lim\limits_{(x,y)\to(0,0)}\frac{R_2(x,y)}{||(x,y)||^2}=0$
The hessian at $(0,0)$, $\frac{(x-y)^2}{2}$, quadratic of the Hessian matrix $\begin{pmatrix}1&-1\\-1&1\end{pmatrix}$ is positive semidefinite so I can guarantee that the point is not a maximum. I can also prove that the point is a local minimum if we approach through the line $x=y$ at which the hessian zeroes. Is this enough to show it is a minimum, i.e, showing that it is through the degenerate direction? If not, is there some clever algebra im missing with the Taylor expression I can use to prove this, or do I have to seek another method?
Thanks in advance