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A real number x when added to its reciprocal give minimum value to the sum when x is
- a)1/2
- b)-1
- c)1
- d)2
Correct answer is option 'C'. Can you explain this answer?
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A real number x when added to its reciprocal give minimum value to the...
Finding the Real Number that Gives Minimum Value to the Sum
Solution:
Let x be the real number. Then, its reciprocal is 1/x.
The sum of x and its reciprocal is x + 1/x.
To find the minimum value of this sum, we can use the concept of the arithmetic mean and geometric mean inequality.
We know that for any two positive numbers a and b, the arithmetic mean is (a+b)/2 and the geometric mean is √(ab).
The arithmetic mean is always greater than or equal to the geometric mean, i.e., (a+b)/2 ≥ √(ab).
Let's apply this inequality to x and 1/x.
The arithmetic mean of x and 1/x is (x + 1/x)/2.
The geometric mean of x and 1/x is √(x * 1/x) = √1 = 1.
By the arithmetic mean and geometric mean inequality, we have:
(x + 1/x)/2 ≥ √(x * 1/x) = 1
Multiplying both sides by 2 gives:
x + 1/x ≥ 2
Therefore, the minimum value of x + 1/x is 2, which is attained when x=1.
Hence, the real number x that gives minimum value to the sum x + 1/x is 1.
Solution:
Let x be the real number. Then, its reciprocal is 1/x.
The sum of x and its reciprocal is x + 1/x.
To find the minimum value of this sum, we can use the concept of the arithmetic mean and geometric mean inequality.
We know that for any two positive numbers a and b, the arithmetic mean is (a+b)/2 and the geometric mean is √(ab).
The arithmetic mean is always greater than or equal to the geometric mean, i.e., (a+b)/2 ≥ √(ab).
Let's apply this inequality to x and 1/x.
The arithmetic mean of x and 1/x is (x + 1/x)/2.
The geometric mean of x and 1/x is √(x * 1/x) = √1 = 1.
By the arithmetic mean and geometric mean inequality, we have:
(x + 1/x)/2 ≥ √(x * 1/x) = 1
Multiplying both sides by 2 gives:
x + 1/x ≥ 2
Therefore, the minimum value of x + 1/x is 2, which is attained when x=1.
Hence, the real number x that gives minimum value to the sum x + 1/x is 1.
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Community Answer
A real number x when added to its reciprocal give minimum value to the...
When x is 1:x + 1/x 》 1+ 1/1 = 2When x is more than 1(i.e. 2)x + 1/x 》 2 + 1/2 = 5/2 = 2.5And the result will keep on increasing when x = 3,4,5,.....nWhen x is less than 1(i.e. 1/2)x + 1/x 》 1/2 + 2 = 5/2 = 2.5And the result will keep on increasing when x = 1/3, 1/4, 1/5,.....n or any no. Less than 1Therefore, minimum value of the sum of a real no. And its reciprocal is derived when x = 1
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A real number x when added to its reciprocal give minimum value to the sum when x isa)1/2b)-1c)1d)2Correct answer is option 'C'. Can you explain this answer?
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A real number x when added to its reciprocal give minimum value to the sum when x isa)1/2b)-1c)1d)2Correct answer is option 'C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A real number x when added to its reciprocal give minimum value to the sum when x isa)1/2b)-1c)1d)2Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A real number x when added to its reciprocal give minimum value to the sum when x isa)1/2b)-1c)1d)2Correct answer is option 'C'. Can you explain this answer?.
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