CAT Question > a, b and c are positive real numbers such th...

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)(p
^{3}- abc) - b)8p
^{3}/27 - c)5p
^{3}/16 - d)9p
^{3}/8

Correct answer is option 'B'. Can you explain this answer?

(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