-256x256.png)
The Arithmetic–Geometric Mean Inequality:
If a ≥ 0, b ≥ 0, then
(a + b)/2 ≥ √(ab) ≥ b.
Equality holds if and only if a = b.
Intro:
1. It is known that a + b = −7, ab = 12. Find:
a) (a + b)²
b) a² + b²
c) (a − b)²
d) a² − ab + b²
e) a³ + b³
2. For a real number x ≠ 0, it is known that (x − 1/x)² = 3. Find all values of:
a) x − 1/x
b) x + 1/x
c) (x + 1/x)²
d) x³ − 1/x³
Inequalities:
3. Prove that for all values of x the following inequalities hold:
a) x² + 2x + 1 ≥ 0
b) 9x² − 6x + 1 ≥ 0
c) x² − 4x + 3 ≥ −1
d) x² + x + 2 ≥ 0
4. Prove for all a and b the inequalities:
a) a² + b² ≥ 2ab
b) (c²a²)/2 + (b²)/(2c²) ≥ ab (c is any nonzero number)
c) (a² + b²)/2 ≥ ((a + b)/2)²
d) (a + b)/2 ≥ 2 / (1/a + 1/b) (for a, b > 0)
5.
a) What is the minimum value of 32x² + 1/(8x²) for x ≠ 0?
b) At which x is the minimum reached?
c) What is the maximum value of 32x + 1/(8x) for negative x?
d) At which x is the maximum reached?
6. Find all pairs of natural numbers x and y satisfying the equation:
a) x² − y² = 13
b) x² − y² = 21
7. Prove that the squares of two natural numbers cannot differ by 14.
8. For any a, b, c > 0 prove the inequalities:
a) a² + b² + c² ≥ ab + bc + ac
b) (a + b)(b + c)(c + a) ≥ 8abc
9.
a) Derive the formula (a + b + c)².
b) It is known that a + b + c = 5 and ab + bc + ac = 5. What can a² + b² + c² be?
c) It is known that x + y + z = 0. Prove that xy + yz + zx ≤ 0.
