# 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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