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Question:

Let a1, a2, a3... be an A.P with a6 = 2. Then the common difference of this A.P., which maximises the product a1a4a5 is:

65

32

85

23

Solution:

Let a be the first term and d be the common difference then,
a + 5d = 2 (1)
f(d) = (2 - 5d)(2 - 2d)(2 - d)
f'(d) = 0 ⇒ d = 2/3, 8/5
f''(d) < 0 ⇒ d = 8/5