Wednesday
July 23, 2014

Homework Help: math

Posted by mathstudent on Thursday, January 4, 2007 at 11:53am.

I'm having trouble understanding one step in a proof of the Cauchy-Schwarz inequality:

u = a non-zero vector
v = another vector
a = <u,u> (so a > 0 by positivity axiom)
b = 2<u,v>
c = <v,v> (so c >= 0 by positivity axiom)
t = any real number

0 <= <tu + v, tu + v> (by positivity axiom)
0 <= t^2<u,u> + 2t<u,v> + <v,v>
0 <= at^2 + bt + c

I understand thus far. But then, the book says, "This inequality implies that the quadratic polynomial at^2 + bt + c has either no real roots or a repeated real root. Therefore its discriminant must satisfy the inequality b^2 - 4ac <= 0"

Why does the inequality imply that the quadratic has no real roots or a repeated real root? Why can't the quadratic have two different roots if b^2 - 4ac > 0?

If there are two different roots, say y1 and y2 such that y1 < y2, then the function would factorize as:


at^2 + bt + c = A(t-y1)(t-y2)

This means that the function will change sign if you let t increase from a value smaller than y1 to a value larger than y1 but still smaller than y2. But a change of sign of this function is precluded because it has to be larger than zero for all t.

makes perfect sense. thanks!

Answer this Question

First Name:
School Subject:
Answer:

Related Questions

geometry - The medial triangle of a triangle ABC is the triangle whose vertices...
maths - Given that vector c and vector d are non zero vectors such that vector c...
Calculus and vectors - Please help/verify the answer. 1. When you add vectors ...
Physics - partscoretotalsubmissions1--10/12--10/13--10/14--10/1----4--The tail ...
Calculus - Dot Product - consider a rhombus ABCD a) find the resultant of vector...
Physics or Math - A force F = 0.8 i + 2.5 j N is applied at the point x = 3.0 m...
Math - vectors - In the product F(vector)=q(V(vector)xB(vector), take q = 4, V(...
Physics - Let vector A = 4i^ + 4j^, vector B = -2i^ - 5j^, and vector F = vector...
Physics - Let vector A = 4i^ + 4j^, vector B = -2i^ - 5j^, and vector F = vector...
Physics - Which of the following is an accurate statement? a) a vector cannot ...

Search
Members