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Question:

The minimum value of the sum of real numbers a⁵, a⁴, 3a³, 1, a⁸ and a¹⁰ with a > 0 is?

6

9

7

8

Solution:

Since, AM ≥ GM-> (1/a⁵ + 1/a⁴ + 1/a³ + 1/a³ + 1/a³ + 1 + a⁸ + a¹⁰)/8 ≥ (1/a⁵ * 1/a⁴ * 1/a³ * 1/a³ * 1/a³ * 1 * a⁸ * a¹⁰)^(1/8)-> (1/a⁵ + 1/a⁴ + 1/a³ + 1/a³ + 1/a³ + 1 + a⁸ + a¹⁰)/8 ≥ (1)^(1/8)-> 1/a⁵ + 1/a⁴ + 1/a³ + 1/a³ + 1/a³ + 1 + a⁸ + a¹⁰ ≥ 8(1)^(1/8)-> 1/a⁵ + 1/a⁴ + 1/a³ + 1/a³ + 1/a³ + 1 + a⁸ + a¹⁰ ≥ 8So, the minimum value is 8