# math

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 + !@#\$%^&as 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!

1. 👍 0
2. 👎 0
3. 👁 106

## Similar Questions

1. ### Precalc

Let x and y be real numbers such that x^2 + y^2 = 1. Find the max value of 2x - 5y. A hint was given to use Cauchy-Schwarz but I'm not sure how to apply it to vectors?

asked by Katie on June 7, 2016
2. ### math

would this be a correct proof of the cauchy-schwartz inequality: abs=absolute value abs(u*v) is less than or equal to abs(u)*abs(v). Then you divide both sides by abs(u)*abs(v) so that you get cos(theta) is less than or equal to

asked by annie on May 14, 2009
3. ### Math

What is wrong with the following proof. You must explain your answer in words. a>3 3a>3(3) 3a-a^2>9-a^2 a(3-a)>(3-a)(3+a) a>3+a 0>3 the problem with the proof is that the fourth step needs the sign to be reverse...but why? i

asked by Wenny on January 26, 2008
4. ### geometry

Construction: The midpoint of a line segment can be constructed. Theorem: If the consecutive midpoints of the sides of a parallelogram are joined in order, then the quadrilateral formed from the midpoints is a parallelogram. Write

asked by carolyn on July 10, 2014
5. ### precal

Prove sin 1 cos  1 cos sin  0 1. Show each step of your proof. 2. Provide written justification for each step of your proof. C. If you use sources, include all in-text citations and references in APA

asked by anoymous on July 13, 2012
6. ### trig

Prove sin 1 cos  1 cos sin  0 1. Show each step of your proof. 2. Provide written justification for each step of your proof. C. If you use sources, include all in-text citations and references in

asked by carolyn on July 10, 2012
7. ### Algebra

Which choice could be the equation of a line parallel to the line represented by this equation? 7x− 12y = −32 A. y = 12/7x - 3 B. y = -7/12x + 5 C. y = 7/12x - 9 D. y = 7x - 12 can someone please give me a step-by-step on how

8. ### Math 9

Hi, I'm having trouble understanding inequality like graphing it on a number line. I don't understand the open circle and close circle. Can someone help explain? Thank you

asked by Lisa on December 30, 2009
9. ### political theory

I don't have an assignment due, but I'm having trouble understanding the themes that are in the Persian Letters by Montesquieu. Has anyone read them? I'm in college. I'm having trouble understanding the Harem sequence, and these

asked by bayley on March 3, 2015