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Prove-that-AM-gt-HM-




Question Number 1268 by 314159 last updated on 18/Jul/15
Prove that AM > HM.
$$\boldsymbol{\mathrm{Prove}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{AM}}\:>\:\boldsymbol{\mathrm{HM}}. \\ $$
Answered by prakash jain last updated on 18/Jul/15
AM=((a+b)/2), HM=((2ab)/(a+b))  AM−HM=((a+b)/2)−((2ab)/(a+b))= (((a−b)^2 )/(2(a+b)))  If a,b>0 then AM≥HM  AM ≥HM is only valid for +ve numbers.  Try: a=−4, b=2
$$\mathrm{AM}=\frac{{a}+{b}}{\mathrm{2}},\:\mathrm{HM}=\frac{\mathrm{2}{ab}}{{a}+{b}} \\ $$$$\mathrm{AM}−\mathrm{HM}=\frac{{a}+{b}}{\mathrm{2}}−\frac{\mathrm{2}{ab}}{{a}+{b}}=\:\frac{\left({a}−{b}\right)^{\mathrm{2}} }{\mathrm{2}\left({a}+{b}\right)} \\ $$$$\mathrm{If}\:{a},{b}>\mathrm{0}\:\mathrm{then}\:\mathrm{AM}\geqslant\mathrm{HM} \\ $$$$\mathrm{AM}\:\geqslant\mathrm{HM}\:\mathrm{is}\:\mathrm{only}\:\mathrm{valid}\:\mathrm{for}\:+\mathrm{ve}\:\mathrm{numbers}. \\ $$$$\mathrm{Try}:\:{a}=−\mathrm{4},\:{b}=\mathrm{2} \\ $$

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