Is \(\displaystyle xy\) (\(\displaystyle \R^2\to \R\)) differentiable? What is the derivative?

Difficulty: 1.

\(\displaystyle f(x,y)=x^2+y^3\), \(\displaystyle g(x,y)=x^2+y^4\). Calculate the first and second differentials at \(\displaystyle (0,0)\).

Difficulty: 1.

Prove that if \(\displaystyle f:\R^2\to\R\) has partial derivative \(\displaystyle D_1f\equiv0\), then \(\displaystyle f\) only depends on \(\displaystyle y\).

Difficulty: 1.

Prove that \(\displaystyle (x,y)\mapsto x/y\) is differentiable (\(\displaystyle y\ne0)\). What is the derivative?

Difficulty: 1.

Write down the second degree Taylor-polynomial of \(\displaystyle xyz\) at \(\displaystyle (1,2,3)\).

Difficulty: 1.

Write down the third degree Taylor-polynomial of \(\displaystyle \sin(x+y)\) at \(\displaystyle (0,0)\).

Difficulty: 1.

Let

\(\displaystyle g(t)=\begin{cases} t^2 & \text{if $t\geq 0$} \\ -t^2 & \text{if $t<0$}\end{cases}\)

At what points is \(\displaystyle f(x,y):=g(x)+g(y)\) differentiable?

Difficulty: 2.

Sketch the level curves of \(\displaystyle f(x,y)=e^{{2x\over x^2+y^2}}\). Given \(\displaystyle (x_0,y_0)\) in which direction does \(\displaystyle f\) grow fastest?

Difficulty: 2.

Let \(\displaystyle F:\R^2\to\R\) be differentiable with derivative \(\displaystyle (f(x,y),g(x,y))\). What is the derivative of \(\displaystyle F(\sin t, \cos t)\)?

Difficulty: 2.

"Let \(\displaystyle f(x,y)=\hbox{log}\sqrt{(x-a)^2+(y-b)^2}\). Show that \(\displaystyle {\partial ^2 f\over \partial x^2}+{\partial^2f \over \partial y^2}=0\).

Difficulty: 2.

\(\displaystyle f:\R^2\to \R\) is smooth. Give a normal vector of the graph of \(\displaystyle z=f(x,y)\) at the point \(\displaystyle (x_0,y_0,f(x_0,y_0))\).

Difficulty: 2.

Prove that \(\displaystyle (x_1,x_2,\ldots,x_n)\mapsto x_1+x_2+\ldots+x_n\) is differentiable. What is its derivative? For a given \(\displaystyle \varepsilon\) find \(\displaystyle \delta\)!

Difficulty: 2.

Prove that \(\displaystyle (x,y)\mapsto x^y\) is continuously differentiable on \(\displaystyle \{ (x,y)\in \R^2\,:\,y>0\}\). What is the derivative?

Difficulty: 2.

Find the derivative of the scalar product of \(\displaystyle n\)-dimensional vectors when viewed as an \(\displaystyle \R^{2n}\to\R\) function.

Difficulty: 2.

Prove that \(\displaystyle (x_1,x_2,\ldots,x_n)\mapsto x_1x_2\ldots x_n\) is differentiable. What is the derivative?

Difficulty: 2.

Let \(\displaystyle f(x,y)=\psi(x-ay)+\phi(x+ay)\), where \(\displaystyle \psi,\phi\) are smooth.

\(\displaystyle {\partial^2 f \over \partial y^2} - a^2 {\partial^2 f\over \partial x^2}=?\)

Difficulty: 2.

At which points is \(\displaystyle ||\ .\ ||_1:=\sum |x_i|\) differentiable?

Difficulty: 3.

Let \(\displaystyle 1<p<\infty\). At which points is the \(\displaystyle ||\ .\ ||_p:=\left(\sum |x_i|^p\right)^{1/p}\) function differentiable?

Difficulty: 3.

\(\displaystyle f(x,y)=\begin{cases} 0 & \text{if $(x,y)=(0,0)$} \\ (x^2+y^2)\sin {1\over \sqrt{x^2+y^2}} & \text{otherwise} \end{cases}\)

is differentiable everywhere but not continuously.

Difficulty: 3.

Let \(\displaystyle g(t)=\sgn(t)\cdot t^2\). Show that \(\displaystyle f(x,y)=g(x)+g(y)\) is everywhere differentiable but is not twice differentiable along the two axes.

Difficulty: 3.

Show that \(\displaystyle (x-y^2)(2x-y^2)\) has no local minimum at \(\displaystyle (0,0)\) even though it has a local minimum along any lines through \(\displaystyle (0,0)\).

Difficulty: 3.

Find the minimum and maximum of \(\displaystyle x^3+x^2-xy\) on \(\displaystyle [0,1]\times[0,1]\).

Difficulty: 3.

Find the maximum and minimum of \(\displaystyle xy\cdot\log(x^2+y^2)\) on \(\displaystyle x^2+y^2\le r\).

Difficulty: 3.

Prove that \(\displaystyle \displaystyle f(x,y)=\frac{x^3}{x^2+y^2}\), \(\displaystyle f(0,0)=0\) has directional derivatives at the origin in all directions. Is there a vector \(\displaystyle a\) such that for all \(\displaystyle v\) unit vector one has \(\displaystyle D_vf(0,0)=a\cdot v\)?

Difficulty: 3.

Find the local extrema of

\(\displaystyle x^2+xy+y^2-3x-3y+5; \qquad x^3y^2(2-x-y). \)

Difficulty: 3.

Find the local extrema of the following functions:

\(\displaystyle x^3+y^3-9xy; \qquad \sin x+\sin y+\sin(x+y) \)

Difficulty: 3.

Prove that if \(\displaystyle f_1,...,f_p:\R\to\R\) are twice differentiable and convex then \(\displaystyle g(x_1,\ldots,x_p)=f_1(x_1)+\ldots+f_p(x_p)\) is also convex.

Difficulty: 3.

What are the local extrema of \(\displaystyle xy+{1\over x}+{1\over y}\)?

Difficulty: 3.

Let

\(\displaystyle f(x,y)=\begin{cases} 0 & \text{if $(x,y)=(0,0)$} \\ xy{x^2-y^2 \over x^2+y^2} & \text{otherwise.} \end{cases}\)

\(\displaystyle {\partial^2 f\over \partial y \partial x}(0,0)=? \qquad {\partial^2 f\over \partial x \partial y}(0,0)=?\)

Difficulty: 4.

Is the function \(\displaystyle (x,y)\mapsto \arcsin {x\over y}\) uniformly continuous?

Difficulty: 4.

Describe those \(\displaystyle f:\R^2\to\R\) for which \(\displaystyle D_1f\equiv D_2f\)?

Difficulty: 4. Hint is provided for this problem.

Prove that if \(\displaystyle f:\R^p\to\R\) is differentiable at \(\displaystyle a\), \(\displaystyle f(a)=0\) and \(\displaystyle f'(a)=0\), then for all bounded \(\displaystyle g:\R^p\to\R\), \(\displaystyle gf\) is differentiable at \(\displaystyle a\).

Difficulty: 4.

For which values of \(\displaystyle \alpha,\beta>0\) is \(\displaystyle |x|^\alpha\cdot|y|^\beta\) twice differentiable at the origin?

Difficulty: 4.

For what \(\displaystyle c\) is

\(\displaystyle f(x,y)=\begin{cases} {|x|^cy \over \sqrt{x^2+y^2}} & \text{if $(x,y)\not= (0,0)$} \\ 0 & \text{if $(x,y) = (0,0)$}\end{cases}\)

differentiable?

Difficulty: 4.

Let \(\displaystyle H\subset\R^{p+q}\), \(\displaystyle a\in\R^p\), \(\displaystyle b\in\R^q\), \(\displaystyle (a,b)\in\INT H\) and \(\displaystyle f:H\to\R\) differentiable at \(\displaystyle (a,b)\) and assume that near \(\displaystyle a\) there is a differentiable function \(\displaystyle \varphi\) to \(\displaystyle \R^q\) such that \(\displaystyle f(x,\varphi(x))=0\). Prove that

\(\displaystyle f_a'(b) \circ \varphi'(a) = -(f^b)'(a). \)

Difficulty: 4.

For \(\displaystyle |x|<1\), \(\displaystyle |y|<1\), \(\displaystyle |z|<1\) let \(\displaystyle u(x,y,z)\) be the real root of

\(\displaystyle (2+x)u^3+(1+y)u-(3+z)=0. \)

Find \(\displaystyle u'(0,0,0)\).

Difficulty: 4.

For \(\displaystyle |x_1-10|<1\), \(\displaystyle |x_2-20|<1\), \(\displaystyle |x_3-30|<1\) let \(\displaystyle u=(u_1,u_2)\) be the root of

\(\displaystyle u_1+u_2=x_1+x_2+x_3-10, \quad u_1u_2=\frac{x_1x_2x_3}{10} \)

closest to \(\displaystyle (30,20)\). Find \(\displaystyle u'(10,20,30)\).

Difficulty: 4.

Given the constraints \(\displaystyle x^2+y^2=1\), \(\displaystyle x^2+z^2=1\) find the largest possible values of \(\displaystyle x\), \(\displaystyle x+y+z\), and \(\displaystyle y+z\).

Difficulty: 4.

Find the maximum of \(\displaystyle xyz\) given the constraints \(\displaystyle x+y+z=5\) and \(\displaystyle x^2+y^2+z^2=9\).

Difficulty: 4.

What is the image of \(\displaystyle x^2+y^2\le1\) under the map \(\displaystyle x^2y^3\log(x^2+y^2)\).

Difficulty: 4.

Find the distance of \(\displaystyle (5,5)\) from the hyperbola \(\displaystyle xy=4\) using Lagrange multiplicators.

Difficulty: 4.

Is the function

\(\displaystyle f(x,y,z)=\begin{cases} \frac{\sin^2x+\sin^2y+\sin^2z}{x^2+y^2+z^2} & (x,y,z)\ne(0,0,0) \\ 1 & x=y=z=0 \\ \end{cases} \)

differentiable at the origin?

Difficulty: 4.

Let \(\displaystyle f:\R^2\to\R\) be the distance of \(\displaystyle (x,y)\) from the interval \(\displaystyle I:=[0,1]\times\{0\}\). At which points is \(\displaystyle f\) differentiable? Twice differentiable?

Difficulty: 5.

Give a function \(\displaystyle g\) whose directional derivatives all exist and vanish at the origin, but

(a) \(\displaystyle g\) is not differentiable at the origin;

(b) not continuous at the origin;

(c) not bounded in any neighborhood of the origin.

Difficulty: 5.

Assume that \(\displaystyle f:\R^2\to\R\) has a second partial derivative \(\displaystyle D_{12}f\) and for all \(\displaystyle a<b\), \(\displaystyle c<d\) we have \(\displaystyle f(a,c)+f(b,d)\ge f(a,d)+f(b,c)\). Show that \(\displaystyle D_{12}\) is non-negative.

Difficulty: 5.

Find the derivative of \(\displaystyle \mathrm{tr}:\R^{n\times n}\to\R\), \(\displaystyle \mathrm{tr}\begin{pmatrix} a_{11} & \dots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \dots & a_{nn} \\ \end{pmatrix}=a_{11}+a_{22}+\ldots+a_{nn}\).

Difficulty: 5. Answer (final result) is provided for this problem.

True or false? If \(\displaystyle f:\R^2\to\R\) is differentiable and for all lines through \(\displaystyle a\) \(\displaystyle f\) has a local minimum at \(\displaystyle a\) along the line, then \(\displaystyle f\) has a local minimum at \(\displaystyle a\).

Difficulty: 5.

Let \(\displaystyle B\) be a real \(\displaystyle q\times r\) matrix. What is the derivative of

\(\displaystyle f(x_1,\ldots,x_{q+r}) = (x_1,\ldots,x_q)M(x_{q+1},\ldots,x_{q+r})^T?\)

Difficulty: 5. Answer (final result) is provided for this problem.

True or false? If \(\displaystyle f:\R^2\to\R\) is differentiable at all points except perhaps at the origin and at the origin it has vanishing directional derivatives in all directions, then \(\displaystyle f\) is differentiable at the origin?

Difficulty: 5.

How many local maximum and minimum places exist for \(\displaystyle (1+e^y)\cos x-ye^y\)?

Difficulty: 5.

Let \(\displaystyle A\) and \(\displaystyle B\) be \(\displaystyle n\times n\) real symmetric matrices where \(\displaystyle \det A\ne0\).

(a) Prove that if \(\displaystyle x\to x^TBx\) has a local extremum at \(\displaystyle x_0\in\R^n\) given the constraint \(\displaystyle x^TAx=1\), then \(\displaystyle x_0\) is an eigenvector of \(\displaystyle A^{-1}B\).

(b) What is the meaning of the eigenvalue corresponding to the eigenvector \(\displaystyle x_0\)?

Difficulty: 5.

Let \(\displaystyle f:\R^3\to\R\) be twice differentiable. Prove that if

\(\displaystyle \left<f'(x,y,z), (x,y,z)\right> \ge 0, \)

holds everywhere, then

\(\displaystyle D_{11}f(0,0,0) + D_{22}f(0,0,0) + D_{33}f(0,0,0) \ge 0. \)

Difficulty: 5.

Assume that \(\displaystyle f:\R^2\to\R\) has a second partial derivative \(\displaystyle D_{12}f\) which is non-negative. Show that if \(\displaystyle a<b\) and \(\displaystyle c<d\), then \(\displaystyle f(a,c)+f(b,d)\ge f(a,d)+f(b,c)\).

Difficulty: 6.

Given \(\displaystyle p_1,\ldots,p_n\) in 3-space we are looking for the plane through the origin for which the sum of the squared distances from the points to the plane is minimal. Let \(\displaystyle v\) be the normal vector of this plane, where \(\displaystyle |v|=1\).

(a) Show that \(\displaystyle v\) is an eigenvector of the matrix \(\displaystyle \sum\limits_{i=1}^n p_ip_i^T\).

(b) What is the geometric meaning of the eigenvalue corresponding to the eigenvector \(\displaystyle v\)?

Difficulty: 6.

Give a function \(\displaystyle f:\R^2\to\R\) for which all directional derivatives exist at \(\displaystyle (0,0)\) but which is not differentiable at \(\displaystyle (0,0)\).

Difficulty: 7.

Assume that \(\displaystyle f:\R^2\to\R\) is differentiable and for all \(\displaystyle x,y\) we have

\(\displaystyle y^2\cdot D_1f(x,y)=x^2\cdot D_2f(x,y). \)

Prove that \(\displaystyle f(x,y)=g(x^3+y^3)\) for some \(\displaystyle g\). Is it necessarily true that the function \(\displaystyle g\) is differentiable at \(\displaystyle 0\)?

Difficulty: 7. Hint is provided for this problem.

Prove that if \(\displaystyle f:\R^2\to\R\) is differentiable and \(\displaystyle D_1f(x,y) = yD_2f(x,y)\) for all \(\displaystyle x,y\), then there is a \(\displaystyle g:\R\to\R\) differentiable function for which \(\displaystyle f(x,y)=g(e^xy)\).

Difficulty: 7.

Prove that if \(\displaystyle H\subset\R^p\) is convex and open and \(\displaystyle f:H\to\R\) is convex, then \(\displaystyle f\) is Lipschitz on all compact subsets of \(\displaystyle H\).

Difficulty: 7.

We know that \(\displaystyle f:\R^2\to\R\) is differentiable and

\(\displaystyle y^2\cdot D_1f(x,y) + x^3\cdot D_2f(x,y) = 0. \)

Prove that \(\displaystyle f(\sqrt2,\sqrt[3]3)=f(0,0)\).

Difficulty: 7.

Prove that if \(\displaystyle D_{12}f\) and \(\displaystyle D_{21}f\) exist in a neighborhood of \(\displaystyle (a,b)\) and they are both continuous at \(\displaystyle (a,b)\) then \(\displaystyle D_{12}f(a,b)=D_{21}f(a,b)\).

Difficulty: 8.

Prove that if \(\displaystyle D_{1}f\), \(\displaystyle D_2f\) and \(\displaystyle D_{12}f\) exist in a neighborhood of \(\displaystyle (a,b)\) and \(\displaystyle D_{12}\) is continuous at \(\displaystyle (a,b)\), then \(\displaystyle D_{21}\) exists and \(\displaystyle D_{12}f(a,b)=D_{21}f(a,b)\). (Schwarz)

Difficulty: 8.

Given \(\displaystyle F:\R^p\to\R\) twice differentiable convex function we are looking for the minimum of \(\displaystyle F\) using the conjugate gradient method: start with \(\displaystyle x_0\) and let

\(\displaystyle x_{n+1} = x_n - c(x_n)\cdot \mathrm{grad}f(x_n), \)

where \(\displaystyle c(x_n)\) is computed from the first and second derivatives of \(\displaystyle f\) at \(\displaystyle x_n\).

(a) What is a good choice for \(\displaystyle c(x_n)\)?

(b) Prove that method works for quadratic forms.

Difficulty: 9.