Following Chapter 2 of Guler.

Class notes (pretty incomplete, to be used just as a study guide), here and here.

**Exercises**:

- Does there exist a differentiable function on unit interval with infinitely many global maxima and and global minima?
- Does there exist a function (on real plane) with infinitely many global maxima but no local minima?
- If a function on real line has a local minimum at \(x\); does it imply that for some \(\epsilon>0\) the function increases on \([x,x+\epsilon)\), decreases on \((x-\epsilon,x]\)?
- How many critical points the function

\[

f(x,y)=xy(x+y-3)(5x+y-5)(x+5y-5)

\]

has on the plane? How many of them are saddles? local maxima? local minima?*Hint: logarithm of a linear function is concave.* - Consider the symmetric operator with

\[

A_{k k+1}=A_{k+1 k}=1,k=1,\ldots,n-1; A_{k k}=2-(-1)^k, k=1,\ldots,n,

\]

and all other entries are zeros.For which \(n\) is the corresponding quadratic form \((Av,v)\) positive definite?

**Homework** (due by midnight of Tuesday, Feb 5th. Submit at Gradescope, course MXX6X4).

- Does there exist a function (on real plane) with infinitely many saddles but no local minima or maxima? Construct, or explain why it is impossible.
- Can a sequence of strict local maxima converge to a global minimum for a differentiable function? Construct, or explain why it is impossible.
- How many critical points the function

\[

f(x,y)=(x^2-1)(y^2-1)(x^2+2xy+y^2-1)

\]

has on the plane? How many of them are saddles? local maxima? local minima? - Find the point on the plane such the the sum of its distances to four points with coordinates \((-2,1),(3,-6),(2,2),(-3,-3)\) is minimal.
- Find the polynomial \(p(z)=z^3+az^2+bz+c\) of degree 3 such that its \(L_2\) norm on the interval \([-1,1]\) is minimal. In other words, minimize

\[

\int_{-1}^1 p(z)^2 dz.

\] - Find minimum and maximum of the quadratic form \(\sum_{k,l=1}^n v_kv_l\) over the unit sphere \(S_1=\{\sum_{k=1}^n v_k^2=1\}\).

Are we allowed to use MATLAB to solve homework problems?

Thanks!

yes, but need to explain what you are doing, besides “there is a magic button out there”

Hello, I have two questions:

1. For problem 2 in homework, what does “a sequence of local maxima” mean?

2. For the last problem in homework, does the first formula mean the inner product of two vector from R^n.

Thank you!

1. It means that you have some local maxima with a commonality in their structure that makes them converge to a global minimum.

2. Note that $v$ is the same vector the function being maximized/minimized not a dot product.

Right,

1. “Sequence of points that are local minima”, – sounds better?

2. \(v_k, k=1,\ldots,n\) are coordinates of the vector. Is that what you’re asking?

Regarding professor’s answer, “1. “Sequence of points that are local minima”, – sounds better?” –> Do we have a sequence of points that are local minima or maxima?

They are local maxima, but they converge to a global minimum.

Maxima, sorry, misspoke.