Inequalities and Optimization Flashcards

For positive reals x and y with xy = 1, what is the minimum value of (x + 1/x)² + (y + 1/y)²?

Answer: 8

Since y = 1/x, both terms equal (x + 1/x)², giving 2(x + 1/x)² ≥ 2·(2)² = 8 by AM-GM.

Among all triangles with perimeter 12, what is the maximum area?

Answer: 4√3

The equilateral triangle maximizes area for a fixed perimeter; with side length 4, area = (√3/4)·16 = 4√3.

For positive reals a and b satisfying a + b = 1, what is the minimum value of a³ + b³?

Answer: 1/4

a³+b³ = (a+b)((a+b)²−3ab) = 1−3ab ≥ 1−3/4 = 1/4, since ab ≤ (a+b)²/4 = 1/4.

What is the maximum value of 2x + y for real x, y satisfying x² + y² ≤ 5?

Answer: 5

By Cauchy-Schwarz, (2x+y)² ≤ (4+1)(x²+y²) ≤ 5·5 = 25, so the maximum is 5.

For non-negative integers m and n with m + n = 10, what is the maximum value of m·n?

Answer: 25

m·n = m(10−m) is maximized at m = 5, giving 5·5 = 25.

Which of the following correctly states the AM-GM inequality for three positive reals x, y, z?

Answer: (x+y+z)/3 ≥ (xyz)^(1/3)

AM-GM states the arithmetic mean (x+y+z)/3 is always ≥ the geometric mean (xyz)^(1/3), with equality iff x=y=z.

What is the minimum value of x² + y² for real numbers x and y satisfying 2x + 3y = 13?

Answer: 13

The minimum squared distance from the origin to the line 2x+3y=13 is 13²/(2²+3²) = 169/13 = 13.