Let $\displaystyle f:\mathbb R^2\to\mathbb R,$ with $\displaystyle f\in\mathcal C^\infty$ such that $\displaystyle f(0,0)=0,$ $\displaystyle \nabla f(0,0)=(0,1),$ $\displaystyle Hf(0,0)=\left( \begin{matrix}1 & 1 \\1 & 2\end{matrix} \right)$ and $\displaystyle F(x,y)=\int_0^{f(x,y)}e^{t^2}\,dt.$

a) Compute the second-order Taylor series for $\displaystyle F$ in $\displaystyle (0,0).$

b) Prove that $\displaystyle F(x,y)=0$ defines $\displaystyle y$ as an implicit function of infinite class, $\displaystyle y=g(x),$ of $\displaystyle x$ around $\displaystyle (0,0).$ Does $\displaystyle g$ have an extrema at $\displaystyle 0$ ?

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