Question

Solve the following question and mark the best possible option.
a, b, c and d are positive real numbers such that a + b + c + d = p, where ‘p’ is a constant. Find the maximum value of (p – a) (p – b) (p – c) (p – d)?

Options

• `((p^4 - abcd))/64`

• `(81p^4)/256`

• `(27p^3)/512`

• `(9p^4)/4`

Answer 1

(p - a), (p - b), (p - c) and (p - d) all are positive number.
We know that, AM  ≥ GM or `[4p - (a + b + c + d)]/4 ≥ [(p - a)(p - b)(p - c)(p - d)]^(1/4) (3p)/4 ≥`

`[(p - a)(p - b)(p - c)(p - d)]^(1/4) ≥ ; ((3p)/4)^4 [(p - a)(p - b)(p - c)(p - d)]^(1/4)`;

`(81p^4)/256 ≥ [(p - a)(p - b)(p - c)(p - d)]`.