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AM GM and HM
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Asked by koushikr063 | 23 May, 2019, 14:39: PM
answered-by-expert Expert Answer
L e t space a space a n d space b space b e space t w o space d i s t i n c t space p o s i t i v e space r e a l space n u m b e r s A M equals fraction numerator a plus b over denominator 2 end fraction comma space G M equals square root of a b end root space a n d space H M equals fraction numerator 2 a b over denominator a plus b end fraction C o n c i d e r space t h e space e q u a t i o n space A x squared minus 2 G x plus H equals 0 rightwards double arrow open parentheses fraction numerator a plus b over denominator 2 end fraction close parentheses x squared minus 2 square root of a b end root x plus fraction numerator 2 a b over denominator a plus b end fraction equals 0 rightwards double arrow open parentheses fraction numerator a plus b over denominator 2 end fraction close parentheses x squared minus square root of a b end root x minus square root of a b end root x plus fraction numerator 2 a b over denominator a plus b end fraction equals 0 rightwards double arrow x open parentheses fraction numerator a plus b over denominator 2 end fraction x minus square root of a b end root close parentheses minus open parentheses fraction numerator 2 square root of a b end root over denominator a plus b end fraction close parentheses open parentheses fraction numerator a plus b over denominator 2 end fraction x minus square root of a b end root close parentheses equals 0 rightwards double arrow open parentheses fraction numerator a plus b over denominator 2 end fraction x minus square root of a b end root close parentheses open parentheses x minus fraction numerator 2 square root of a b end root over denominator a plus b end fraction close parentheses equals 0 rightwards double arrow x equals fraction numerator 2 square root of a b end root over denominator a plus b end fraction N o w comma space a comma b element of straight real numbers to the power of plus space a n d space a not equal to b rightwards double arrow a plus b greater than 2 square root of a b end root rightwards double arrow fraction numerator 2 square root of a b end root over denominator a plus b end fraction less than 1 space a n d space fraction numerator 2 square root of a b end root over denominator a plus b end fraction greater than 0 T h e r e f o r e comma space 0 less than x less than 1 space rightwards double arrow x element of left parenthesis 0 comma 1 right parenthesis H e n c e comma space r o o t s space o f space t h e space g i v e n space q u a d r a t i c space e q u a t i o n space l i e s space i n space t h e space i n t e r v a l space left parenthesis 0 comma 1 right parenthesis. space
Answered by Renu Varma | 23 May, 2019, 17:44: PM

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