Math Help - Second-order Taylor series and Implicit Function Theorem

1. Second-order Taylor series and Implicit Function Theorem

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

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

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

Spoiler:

a) Second-order Taylor series is $F(x_0+h)=F(x_0)+\nabla F(x_0)\cdot h+\frac12h^tH\big(F(x_0)\big)h+R_2(x_0,h),$ so for $(0,0)$ we have $F(h_1,h_2)=\nabla F(0,0)\cdot h+\frac12h^tH\big(F(0,0)\big)h+R_2((0,0),h),$ but I don't know how to compute $\nabla F$ nor $H.$

b) This case is strange to me to apply the IFT, what are the steps?

2. Re: Second-order Taylor series and Implicit Function Theorem

We have $\partial_x F(x,y)=\partial f_x(x,y)e^{f(x,y)^2}$ and $\partial_y F(x,y)=\partial f_y(x,y)e^{f(x,y)^2}$, so you can compute $\nabla F$.

3. Re: Second-order Taylor series and Implicit Function Theorem

I don't know how to apply well the implicit function theorem for part b), can you give me a hand?