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

  1. #1
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    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?
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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 \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.
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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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