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Question Number 215883 by CrispyXYZ last updated on 20/Jan/25

$$\mathrm{Given}m>0,n>0,m+n=\sqrt{a}.$$$$\mathrm{Find}\mathrm{the}\mathrm{range}\mathrm{of}a\mathrm{such}\mathrm{that}$$$$‘‘(m+\frac{1}{m})(n+\frac{1}{n})\mathrm{gets}\mathrm{its}\mathrm{minimum}\mathrm{iff}m=n^{″}.$$

Commented by MathematicalUser2357 last updated on 21/Jan/25

$$‘‘{iff}^{″}?doesitmeanifh.cc?$$

Commented by CrispyXYZ last updated on 21/Jan/25

$$‘‘\mathrm{if}\mathrm{and}\mathrm{only}{\mathrm{if}}^{″}.$$

Answered by Eze120 last updated on 23/Jan/25

$$m=n$$$$(m+m^{-1})(m+m^{-1})=m^2+{\left(m^{-1}\right)}^2$$$$2m=\sqrt{a}$$$$2m^2=a$$$$m^2=\frac{a}{2}$$$${\left(m^{-1}\right)}^2=\frac{1}{m^2}=\frac{1}{a/2}$$$$\frac{a}{2}+\frac{1}{a/2}=\left(1/2\right)a+\frac{2}{a}$$$$PD:{It}^{\prime }saguess,{it}^{\prime }snotaccurate.$$

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