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