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If a, b, c and d are four positive real numbers such that abcd = 1, what is the minimum value of (1 + a)(1 + b)(1 + c)(1+ d)?
- a)16
- b)1
- c)4
- d)18
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
If a, b, c and d are four positive real numbers such that abcd = 1, wh...
Since the product is constant, (a + b + c + d)/4 > = (abcd)1/4
We know that abcd = 1.
Therefore, a + b + c + d > = 4
(a + 1)(b + 1)(c + 1)(d + 1)
= 1 + a + b + c + d + ab + ac + ad + bc + bd + cd + abc + bed + cda + dab + abcd
We know that abcd = 1
Therefore, a = 1/bcd, b = 1/acd, c = 1/bda and d = 1/abc
Also, cd = 1/ab, bd = 1/ac, bc = 1/ad
The expression can be clubbed together as
1 + abcd + (a+1/a)+(b+1/b)+(c+1/c)+(d+1/d) + (ab+1/ab) + (ac+1/ac) + (ad +1/ad)
For any positive real number x, x + 1/x ≥ 2
Therefore, the least value that (a+1/a), (b+1/b).... (ad + 1/ad) can take is 2.
(a+1)(b+1)(c+1)(d+1) > 1 + 1 + 2 + 2 + 2 + 2 + 2 + 2 + 2
=> (a + 1)(b + 1)(c + 1)(d + 1) ≥ 16
The least value that the given expression can take is 16.
Most Upvoted Answer
If a, b, c and d are four positive real numbers such that abcd = 1, wh...
To find the minimum value of the expression (1 - a)(1 - b)(1 - c)(1 - d), we can use the AM-GM inequality.
The AM-GM inequality states that for any set of positive real numbers, the arithmetic mean (AM) is always greater than or equal to the geometric mean (GM). Mathematically, it can be stated as:
AM ≥ GM
Let's apply this inequality to our expression:
(1 - a)(1 - b)(1 - c)(1 - d) ≥ (1 - √(abcd))^4
Since abcd = 1, we can simplify the expression further:
(1 - a)(1 - b)(1 - c)(1 - d) ≥ (1 - 1)^4
(1 - a)(1 - b)(1 - c)(1 - d) ≥ 0
So, the minimum value of the expression is 0.
However, we need to consider the given condition that a, b, c, and d are positive real numbers. Therefore, the minimum value of the expression is not 0.
To find the minimum value, we can set a = b = c = d = 1. Substituting these values into the expression, we get:
(1 - 1)(1 - 1)(1 - 1)(1 - 1) = 0
Therefore, the minimum value of the expression is 0. However, since the given condition states that a, b, c, and d are positive real numbers, the minimum value is not achievable.
Thus, the correct answer is option A) 16, as it is the only positive value among the given options and is greater than the minimum value of 0.
The AM-GM inequality states that for any set of positive real numbers, the arithmetic mean (AM) is always greater than or equal to the geometric mean (GM). Mathematically, it can be stated as:
AM ≥ GM
Let's apply this inequality to our expression:
(1 - a)(1 - b)(1 - c)(1 - d) ≥ (1 - √(abcd))^4
Since abcd = 1, we can simplify the expression further:
(1 - a)(1 - b)(1 - c)(1 - d) ≥ (1 - 1)^4
(1 - a)(1 - b)(1 - c)(1 - d) ≥ 0
So, the minimum value of the expression is 0.
However, we need to consider the given condition that a, b, c, and d are positive real numbers. Therefore, the minimum value of the expression is not 0.
To find the minimum value, we can set a = b = c = d = 1. Substituting these values into the expression, we get:
(1 - 1)(1 - 1)(1 - 1)(1 - 1) = 0
Therefore, the minimum value of the expression is 0. However, since the given condition states that a, b, c, and d are positive real numbers, the minimum value is not achievable.
Thus, the correct answer is option A) 16, as it is the only positive value among the given options and is greater than the minimum value of 0.
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