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If x>0, prove that x+1/x is more than equal to zero its ques. of sequence and series
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Proof that x + 1/x is greater than or equal to zero for x > 0
The expression x + 1/x can be rewritten as (x^2 + 1)/x. To prove that this expression is greater than or equal to zero when x is greater than 0, we can use the AM-GM inequality.
AM-GM Inequality
The Arithmetic Mean-Geometric Mean (AM-GM) inequality states that for any non-negative real numbers a and b, the arithmetic mean is greater than or equal to the geometric mean, i.e., (a + b)/2 ≥ √(ab).
Applying AM-GM Inequality
Let a = x^2 and b = 1. By AM-GM inequality, we have:
(x^2 + 1)/2 ≥ √(x^2 * 1)
(x^2 + 1)/2 ≥ x
Conclusion
Since x is greater than 0, the expression (x^2 + 1)/x is greater than or equal to 2x/x = 2, which means x + 1/x is greater than or equal to 2. Therefore, x + 1/x is always greater than or equal to 2 when x is greater than 0, proving that it is greater than or equal to zero.
The expression x + 1/x can be rewritten as (x^2 + 1)/x. To prove that this expression is greater than or equal to zero when x is greater than 0, we can use the AM-GM inequality.
AM-GM Inequality
The Arithmetic Mean-Geometric Mean (AM-GM) inequality states that for any non-negative real numbers a and b, the arithmetic mean is greater than or equal to the geometric mean, i.e., (a + b)/2 ≥ √(ab).
Applying AM-GM Inequality
Let a = x^2 and b = 1. By AM-GM inequality, we have:
(x^2 + 1)/2 ≥ √(x^2 * 1)
(x^2 + 1)/2 ≥ x
Conclusion
Since x is greater than 0, the expression (x^2 + 1)/x is greater than or equal to 2x/x = 2, which means x + 1/x is greater than or equal to 2. Therefore, x + 1/x is always greater than or equal to 2 when x is greater than 0, proving that it is greater than or equal to zero.
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If x>0, prove that x+1/x is more than equal to zero its ques. of sequence and series for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If x>0, prove that x+1/x is more than equal to zero its ques. of sequence and series covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If x>0, prove that x+1/x is more than equal to zero its ques. of sequence and series.
If x>0, prove that x+1/x is more than equal to zero its ques. of sequence and series for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If x>0, prove that x+1/x is more than equal to zero its ques. of sequence and series covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If x>0, prove that x+1/x is more than equal to zero its ques. of sequence and series.
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