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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 (1/a) + (1/b) + (1/c) = (1/(18)) , find minimum value of a.

$${Given}\:{a},{b}\:{and}\:{c}\:{are}\:{real}\:{numbers}\:{and}\:{a}<{b}<{c}. \\ $$ $${If}\:\frac{\mathrm{1}}{{a}}\:+\:\frac{\mathrm{1}}{{b}}\:+\:\frac{\mathrm{1}}{{c}}\:=\:\frac{\mathrm{1}}{\mathrm{18}}\:,\:{find}\:{minimum}\:{value}\:{of}\:{a}. \\ $$

Commented bymr W last updated on 27/Feb/21

no minimum for a exists.

$${no}\:{minimum}\:{for}\:{a}\:{exists}. \\ $$

Commented bymr W last updated on 27/Feb/21

a=(1/((1/(18))−((1/b)+(1/c)))) →18 if b, c →+∞  a=(1/((1/(18))−((1/b)+(1/c)))) →−∞ if b, c →0^+

$${a}=\frac{\mathrm{1}}{\frac{\mathrm{1}}{\mathrm{18}}−\left(\frac{\mathrm{1}}{{b}}+\frac{\mathrm{1}}{{c}}\right)}\:\rightarrow\mathrm{18}\:{if}\:{b},\:{c}\:\rightarrow+\infty \\ $$ $${a}=\frac{\mathrm{1}}{\frac{\mathrm{1}}{\mathrm{18}}−\left(\frac{\mathrm{1}}{{b}}+\frac{\mathrm{1}}{{c}}\right)}\:\rightarrow−\infty\:{if}\:{b},\:{c}\:\rightarrow\mathrm{0}^{+} \\ $$

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

$$\:\mathrm{AM}−\mathrm{GM} \\ $$ $$\:\frac{\frac{\mathrm{1}}{\mathrm{a}}+\frac{\mathrm{1}}{\mathrm{b}}+\frac{\mathrm{1}}{\mathrm{c}}}{\mathrm{3}\:}\:\geqslant\:\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{\mathrm{abc}}} \\ $$ $$\:\frac{\left[\:\frac{\mathrm{1}}{\mathrm{18}}\:\right]}{\mathrm{3}}\:\geqslant\:\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{\mathrm{abc}}}\:\Rightarrow\:\frac{\mathrm{1}}{\mathrm{54}}\:\geqslant\:\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{\mathrm{abc}}} \\ $$ $$\frac{\mathrm{1}}{\mathrm{54}^{\mathrm{3}} }\:\geqslant\:\frac{\mathrm{1}}{\mathrm{abc}}\:,\:\mathrm{it}\:\mathrm{hold}\:\mathrm{for}\:\mathrm{a}=\mathrm{b}=\mathrm{c} \\ $$ $$\mathrm{we}\:\mathrm{get}\:\mathrm{a}\:=\:\mathrm{54}\: \\ $$

Commented bymr W last updated on 27/Feb/21

what do you get if b=c=1000000?  what do you get if b=c=1/1000000?

$${what}\:{do}\:{you}\:{get}\:{if}\:{b}={c}=\mathrm{1000000}? \\ $$ $${what}\:{do}\:{you}\:{get}\:{if}\:{b}={c}=\mathrm{1}/\mathrm{1000000}? \\ $$

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