Proving simple inequalities
Answers are for parents and teachers.
1
Prove a² + 4 ≥ 4a.
a² + 4 − 4a = (a − 2)² ≥ 0.
2
Find the smallest value of x + 16/x for x > 0.
8
3
Prove (a + b)(1/a + 1/b) ≥ 4 for a, b > 0.
Expanding gives 2 + a/b + b/a ≥ 2 + 2 = 4.
4
Compute the arithmetic and geometric means of 2 and 8 and compare them.
arithmetic mean 5, geometric mean 4; 5 ≥ 4
5
Prove x + 9/x ≥ 6 (x > 0) by assuming the opposite.
If x + 9/x < 6, multiplying by x > 0 gives x² − 6x + 9 < 0, i.e. (x − 3)² < 0, which is impossible.