Q5. Find the least value of 4x + 1/x for x > 0.
4x × 1/x = 4, so (4x + 1/x)/2 ≥ √4 = 2 and 4x + 1/x ≥ 4. Equality needs 4x = 1/x, that is x = 1/2. The least value is 4.
Q6. For positive a and b, show that (a + b)(1/a + 1/b) ≥ 4.
a + b ≥ 2√(ab) and 1/a + 1/b ≥ 2/√(ab). Multiplying the two (both sides positive) gives a product of at least 4, with equality when a = b.
Q7. Two positive numbers add up to 10. What is the greatest possible value of their product?
A = 5, and G ≤ A gives √(ab) ≤ 5, so ab ≤ 25, reached when both numbers are 5.
Q8. Insert one geometric mean and one arithmetic mean between 9 and 16, and compare them.
G = √144 = 12 and A = 25/2 = 12.5. A > G, as the numbers are different; 9, 12, 16 is a G.P. and 9, 12.5, 16 is an A.P.