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If the sum of any two positive quantities is constant, then their product is maximum when the two quantities
- a)Are reciprocals of each other
- b)Are a number and its root
- c)Are equal to each other
- d)Are not equal to each other
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
If the sum of any two positive quantities is constant, then their prod...
If the sum of two quantities is constant, then their product is maximum when they are equal.
Alternative Method:
Let X + Y = K
(X + Y)2 - (X - Y)2 = 4XY
K2 - (X - Y)2 = 4XY
⇒ 4XY < K2 (except when X - Y = 0)
Or,
X = Y
⇒ 4XY is maximum when X = Y.
⇒ XY is maximum when X = Y = (K/2)
Thus, the maximum value of XY = (K2/4)
Alternative Method:
Let X + Y = K
(X + Y)2 - (X - Y)2 = 4XY
K2 - (X - Y)2 = 4XY
⇒ 4XY < K2 (except when X - Y = 0)
Or,
X = Y
⇒ 4XY is maximum when X = Y.
⇒ XY is maximum when X = Y = (K/2)
Thus, the maximum value of XY = (K2/4)
Most Upvoted Answer
If the sum of any two positive quantities is constant, then their prod...
Explanation:
To find the maximum product of two positive quantities when their sum is constant, let's assume the two quantities as x and y. The given condition states that x + y = constant.
1. Applying AM-GM inequality:
To maximize the product of x and y, we can use the AM-GM inequality, which states that the arithmetic mean of a set of non-negative numbers is always greater than or equal to their geometric mean.
AM-GM inequality: (x + y)/2 ≥ √(xy)
Since x + y is constant, the right side of the inequality is also constant. To maximize the product xy, we need to maximize the value of √(xy). This can be achieved when the equality holds in the inequality.
2. Maximizing the geometric mean:
We can maximize the geometric mean √(xy) by making x and y equal to each other. Let's assume x = y = k, where k is a positive constant.
Substituting these values into the given condition, we get:
2k = constant
k = constant/2
Therefore, when x = y = constant/2, the product xy is maximized.
3. Proof that other options are not correct:
a) If x and y are reciprocals of each other, then their sum will not be constant, as the sum of reciprocals depends on the values of x and y. Hence, option A is incorrect.
b) If x is a number and y is its root, then their sum will not be constant, as the value of the root depends on the value of x. Hence, option B is incorrect.
d) If x and y are not equal to each other, then their sum will not be constant, as it depends on the specific values of x and y. Hence, option D is incorrect.
Therefore, the correct answer is option C - when x and y are equal to each other.
To find the maximum product of two positive quantities when their sum is constant, let's assume the two quantities as x and y. The given condition states that x + y = constant.
1. Applying AM-GM inequality:
To maximize the product of x and y, we can use the AM-GM inequality, which states that the arithmetic mean of a set of non-negative numbers is always greater than or equal to their geometric mean.
AM-GM inequality: (x + y)/2 ≥ √(xy)
Since x + y is constant, the right side of the inequality is also constant. To maximize the product xy, we need to maximize the value of √(xy). This can be achieved when the equality holds in the inequality.
2. Maximizing the geometric mean:
We can maximize the geometric mean √(xy) by making x and y equal to each other. Let's assume x = y = k, where k is a positive constant.
Substituting these values into the given condition, we get:
2k = constant
k = constant/2
Therefore, when x = y = constant/2, the product xy is maximized.
3. Proof that other options are not correct:
a) If x and y are reciprocals of each other, then their sum will not be constant, as the sum of reciprocals depends on the values of x and y. Hence, option A is incorrect.
b) If x is a number and y is its root, then their sum will not be constant, as the value of the root depends on the value of x. Hence, option B is incorrect.
d) If x and y are not equal to each other, then their sum will not be constant, as it depends on the specific values of x and y. Hence, option D is incorrect.
Therefore, the correct answer is option C - when x and y are equal to each other.
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If the sum of any two positive quantities is constant, then their product is maximum when the two quantitiesa)Are reciprocals of each otherb)Are a number and its rootc)Are equal to each otherd)Are not equal to each otherCorrect answer is option 'C'. Can you explain this answer?
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