Cauchy-Schwarz inequality.

Cauchy-Schwarz inequality offers a quick solution to certain inequality problems. I myself came across at least 1 such question in last year's CAT itself, which, fortunately, having known Cauchy-Schwarz, I could solve in a matter of seconds.

Cauchy-Schwarz states that:

(x₁² + x₂² + x₃²)(y₁² + y₂² + y₃²) ≥ (x₁y₁ + x₂y₂ + x₃y₃)², for real x_i and y_j

Without wasting anytime, we will discuss some relevant problems.

For further reference on Cauchy-Schwarz inequality, follow this link.

Problem 1:

If a, b, c ,d are real numbers with a² + b² + c² + d² = 100, then what is the maximum value of 2a + 3b + 6c + 24d.

Using Cauchy-Schwarz, we can say

(a² + b² + c² + d²)(2² + 3² + 6² + 24²) ≥ (2a + 3b + 6c + 24d)²

(100)(625) ≥ (2a + 3b + 6c + 24d)²

So, 2a + 3b + 6c + 24d ≤ 250

Problem 2:

Find the least value of x² + 4y² + 9z², if x + y + z = 14

{x² + (2y)² + (3z)²}{1 + (1/2)² + (1/3)²} ≥ {x + 2y(1/2) + 3z(1/3)}² 

=> (x² + 4y² + 9z²) ≥ (x + y + z)²/{1 + (1/2) + (1/3)}²

=> (x² + 4y² + 9z²)   ≥ 196/(49/36)

=> (x² + 4y² + 9z²)   ≥ 144

Problem 3:

If x² + y² - 6x + 4y = 4, find the maximum value of 3x + 4y.

Given equation is: (x-3)² + (y+2)² = 9

Using Cauchy Schwarz, we get

[(x - 3)² + (y + 2)²][3² + 4²] ≥ [3(x - 3) + 4(y + 2]²

9 * 25  ≥  (3x + 4y -1)²

3x + 4y -1 ≤ 15

3x + 4y ≤ 16

Regards & Best Wishes,

-Angadbir