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a, b and c are positive real numbers such that a + b + c = p, where 'p' is a constant. Find the maximum value of (p - a) (p - b)(p - c).
  • a)
    (p3 - abc)
  • b)
    8p3/27
  • c)
    5p3/16
  • d)
    9p3/8
Correct answer is option 'B'. Can you explain this answer?

Answers

(p − a), (p − b) and (p − c) all are positive numbers.
∴ AM ≥ GM
Hence (b) is the correct answer.
Alternative Method:
(p − a) + (p − b) + (p − c) = 2p(constant)
⇒ (p − a)⋅(p − b)⋅(p − c) will be maximum when
p − a = p − b = p − c = 2p/3
Maximum value of the product

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(p − a), (p − b) and (p − c) all are positive numbers.∴ AM ≥ GMHence (b) is the correct answer.Alternative Method:(p − a) + (p − b) + (p − c) = 2p(constant)⇒ (p − a)⋅(p − b)⋅(p − c) will be maximum whenp − a = p − b = p − c = 2p/3Maximum value of the product