# Cauchy-Schwarz Inequality By Engr. Roy Roque Rivera Jr. Part 1

Our star of the day is another gifted man. I am actually his number 1 fan. He graduated as valedictorian during high school at Negros Occidental High School. He took up B.S. Chemical Engineering at the University of St. La Salle, Bacolod City. December 3, 2013, he took up the ChE board examination and ranked 6th out of 777 takers and 426 passers. What made him different from other takers was that he just did a SELF-REVIEW. One of our friends told me that he is the guy who always has an answer. He is currently working as a Chemical Engineer at Coral Bay Nickel Corporation.

As a quizzer since grade school, here are just a few glimpse of his achievements.

**College:**

- 6
^{th}Placer Chemical Engineering Board Examinations - Champion, 2011 Visayas – Mindanao Wide MTAP – Tertiary Level (Team and Individual Category)
- 2
^{nd}Place, Philippine Statistics Quiz (Regional Level) - Champion, 2010 Council of Engineering and Architectural Schools (CEAS) – Visayas Math and Science Quiz Region VI (Team and Individual Category)
- 1
^{st}Runner Up, 2011 Council of Engineering and Architectural Schools (CEAS) – Visayas Math and Science Quiz Region VI (2012)

**High School**

- National Finalist, 2007 Metrobank MTAP DepEd Mathematics Challenge (Team and Individual Category)
- Champion, 2007 Metrobank MTAP DepEd Mathematics Challenge Regional Level (Team and Individual Category)

Here is the first part of the article he shared. This is the conventional Cauchy – Schwarz Inequality. The second part of the article is the application of the same inequality in Engel Form which has a huge application in proving hard problems in inequalities.

**1. The Cauchy – Schwarz Inequality**

1.1 Cauchy – Schwarz Inequality

**Theorem: **Let and be real numbers for then the following inequality holds

**Proof: **Let and . Consider

From AM-GM Inequality, we know that (1)^{
}

Or

(1)AM-GM Inequality states that for any positive real numbers the inequality

always holds. Equality is attained if and only if

Therefore, (1) becomes

But and ; hence,

It follows that

Or

Which was to be demonstrated

Equality occurs if and only if

For some constant

**1.2 Cauchy- Schwarz Inequality in Engel Form**

Theorem ( Cauchy-Schwarz Inequality in Engel Form) : Suppose that with for then

(2)

**Proof: **The proof of (2) directly follows from the Cauchy-Schwarz Inequality. From Cauchy- Schwarz Inequality, we have

Hence,

Which was to be demonstrated

Equality occurs if and only if

For some constant

2. **Illustrations and Examples**

**2.1 Elementary Applications**

**Problem 1: **What is the minimum value of given that

**Solution 1: (Method of Lagrange Multipliers): **The method of Lagrange Multipliers ^{2} involves finding the values of and such that the system of equations hold

Now,

And

We have

Hence, we have the following system of equations to solve

Now,

and

Thus,

^{2}The method of Lagrange Multipliers involves finding the extreme values of the function given the constrant

Solving for ʎ gives . Now, we have

Thus, min

**Solution 2 (Cauchy-Schwarz Inequality): **Observe that can be written as . Now, by Cauchy –Schwarz Inequality, we have

or

Solving for gives

Hence, min

**Problem 2: **What is the minimum distance between the ellipse and the line . The ellipse and the plane do not intersect

**Solution: **Let be a point on the ellipse , and be a point on the line . Also, let the distance between and be equal to ; hence,

Now, consider the product

By Cauchy- Schwarz Inequality, we have

Or equivalently

Note that because is one the line , it follows that . Also since the ellipse and the plane do not intersect

Now, consider the product

By Cauchy-Schwarz Inequality, the product becomes

Then

or

Observe that in order to minimize the value of d, the value of must be maximized; hence, take . Therefore, it follows that the minimum distance between the ellipse and the line is equal^{3}

**Problem 3: **Find the maximum value of on the ellipsoid .

**Solution:** We use Cauchy-Schwarz Inequality

or

Thus,

Or

Therefore, the maximum value of is

**Problem 4: **Find the maximum value of on the ellipsoid

**Solution: **By Cauchy – Schwarz Inequality, we have

Or

Thus,

or

Therefore, the maximum value of is

One can use the Method of Lagrange Multipliers and get the same result.

**Problem 5 (R. Guadalupe): **Let such that

Find the maximum value of *a*.

**Solution: **We write the equation as

and

Now, by Cauchy – Schwarz Inequlity we have . Hence,

Which can be simplified into

It follows that

Thus, the maximum value of *a* must be

** **

**Problem 6 (United States of America Mathematical Olympiad):** Let such that

What is the maximum value of *e*?

Solution:

By Cauchy-Schwarz Inequality, we know that

Or equivalently

Or

Which implies that . Thus, max