 Solution

The implicit function theorem applies when $D_{2}F(a,b)\ne 0$. After shrinking the neighbourhood, this partial derivative stays nonzero along the implicit graph. Differentiating its defining identity givesFor the specified smooth function, $F_x=y+\cos (x+y)$ and $F_y=x+\cos (x+y)$, so both equal one at zero. In particular the theorem applies and $f^{\prime }(0)=-1$. Differentiate the identity a second time:At zero, $F_{xx}=F_{yy}=0$ and $F_{xy}=1$. Therefore