Thread: Second-order Taylor series and Implicit Function Theorem

1. Second-order Taylor series and Implicit Function Theorem

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 $\displaystyle 0$ ?

Spoiler:

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

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

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?