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Thread: Second-order Taylor series and Implicit Function Theorem

  1. #1
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    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?
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  2. #2
    Super Member girdav's Avatar
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    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 $\displaystyle \nabla F$.
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  3. #3
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    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?
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