Problem Set 8

  1. Easy
    Define \( f: \mathbb R^2 \to \mathbb R\) by
    \( f(x,y) = \left\{ \begin{matrix} \frac{x^3y-xy^3}{x^2+y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{matrix} \right. \) Compute the mixed partial derivatives \( \partial_{xy} f, \partial_{yx}f \). Why does this not contradict Clairut's theorem?

  2. Easy
    For each of the following functions \(f: \mathbb R^n \to \mathbb R\), compute the Hessian \(Hf(\mathbf x)\).
    1. \(f(x,y) = e^x \cos(y) \)
    2. \(f(x,y) = e^{x^2+y^2} \)
    3. \(f(x,y) = \frac{x}{1+y^2} \)
    4. \(f(x,y,z) = xy\log(2+\cos(z)) \)
    5. \(f(x,y,z,w) = xyz + yzw + zwx \)
    6. \(f(x,y)=\begin{cases} \frac{x^4+y^4}{x^2+y^2} & (x,y)\neq (0,0) \\ 0 & (x,y)=(0,0) \end{cases}\)

  3. Easy
    Let \( f: \mathbb R^n \to \mathbb R \) be a \(C^2\) function. Show that the Hessian matrix \(Hf(\mathbf x) = \left[ \partial_{x_i,x_j} f (\mathbf x) \right] \) is symmetric; that is, \( Hf(\mathbf x)^T = Hf(\mathbf x) \).

  4. Medium
    Suppose that \(f(x,y,z,t)\), \(x(t)\), \(y(x,t,s)\), and \(z(y,x)\). Determine the partial derivatives \( \partial_{ss} f, \partial_{tt}f, \partial_{st} f \).

  5. Medium
    Suppose that \(f : \mathbb R^n \to \mathbb R \) is a \(C^2 \) function, and
    \( \displaystyle Hf(\mathbf x) = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{pmatrix} \)
    is a constant matrix for all \(\mathbf x \in \mathbb R^n\). Determine, up to a linear function, the function \(f\).

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