# ECE 490, week of Jan 21

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 $$S_1=\{\sum_{k=1}^n v_k^2=1\}$$.

Solutions to HW1.

### 8 Responses to ECE 490, week of Jan 21

1. Yiqing Xie January 31, 2019 at 6:37 pm #

Are we allowed to use MATLAB to solve homework problems?
Thanks!

• Yiqing Xie January 31, 2019 at 7:32 pm #

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

2. Erchi Wang February 4, 2019 at 9:28 am #

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!

• Khaled Alshehri February 4, 2019 at 3:50 pm #

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.

• yuliy February 4, 2019 at 4:06 pm #

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?

3. Hakan Tekgul February 4, 2019 at 5:00 pm #

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?

• Khaled Alshehri February 4, 2019 at 5:14 pm #

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

• yuliy February 5, 2019 at 7:57 pm #

Maxima, sorry, misspoke.