Question Number 134069 by john_santu last updated on 27/Feb/21
$${Given}\:{a},{b}\:{and}\:{c}\:{are}\:{real}\:{numbers}\:{and}\:{a}<{b}<{c}. \\ $$$${If}\:\frac{\mathrm{1}}{{a}}\:+\:\frac{\mathrm{1}}{{b}}\:+\:\frac{\mathrm{1}}{{c}}\:=\:\frac{\mathrm{1}}{\mathrm{18}}\:,\:{find}\:{minimum}\:{value}\:{of}\:{a}. \\ $$
Commented by mr W last updated on 27/Feb/21
$${no}\:{minimum}\:{for}\:{a}\:{exists}. \\ $$
Commented by mr W last updated on 27/Feb/21
$${a}=\frac{\mathrm{1}}{\frac{\mathrm{1}}{\mathrm{18}}−\left(\frac{\mathrm{1}}{{b}}+\frac{\mathrm{1}}{{c}}\right)}\:\rightarrow\mathrm{18}\:{if}\:{b},\:{c}\:\rightarrow+\infty \\ $$$${a}=\frac{\mathrm{1}}{\frac{\mathrm{1}}{\mathrm{18}}−\left(\frac{\mathrm{1}}{{b}}+\frac{\mathrm{1}}{{c}}\right)}\:\rightarrow−\infty\:{if}\:{b},\:{c}\:\rightarrow\mathrm{0}^{+} \\ $$
Answered by bramlexs22 last updated on 27/Feb/21
![AM−GM (((1/a)+(1/b)+(1/c))/(3 )) ≥ ((1/(abc)))^(1/3) (([ (1/(18)) ])/3) ≥ ((1/(abc)))^(1/3) ⇒ (1/(54)) ≥ ((1/(abc)))^(1/3) (1/(54^3 )) ≥ (1/(abc)) , it hold for a=b=c we get a = 54](https://www.tinkutara.com/question/Q134073.png)
$$\:\mathrm{AM}−\mathrm{GM} \\ $$$$\:\frac{\frac{\mathrm{1}}{\mathrm{a}}+\frac{\mathrm{1}}{\mathrm{b}}+\frac{\mathrm{1}}{\mathrm{c}}}{\mathrm{3}\:}\:\geqslant\:\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{\mathrm{abc}}} \\ $$$$\:\frac{\left[\:\frac{\mathrm{1}}{\mathrm{18}}\:\right]}{\mathrm{3}}\:\geqslant\:\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{\mathrm{abc}}}\:\Rightarrow\:\frac{\mathrm{1}}{\mathrm{54}}\:\geqslant\:\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{\mathrm{abc}}} \\ $$$$\frac{\mathrm{1}}{\mathrm{54}^{\mathrm{3}} }\:\geqslant\:\frac{\mathrm{1}}{\mathrm{abc}}\:,\:\mathrm{it}\:\mathrm{hold}\:\mathrm{for}\:\mathrm{a}=\mathrm{b}=\mathrm{c} \\ $$$$\mathrm{we}\:\mathrm{get}\:\mathrm{a}\:=\:\mathrm{54}\: \\ $$
Commented by mr W last updated on 27/Feb/21
$${what}\:{do}\:{you}\:{get}\:{if}\:{b}={c}=\mathrm{1000000}? \\ $$$${what}\:{do}\:{you}\:{get}\:{if}\:{b}={c}=\mathrm{1}/\mathrm{1000000}? \\ $$