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Question Number 212992 by CrispyXYZ last updated on 28/Oct/24

$$a>0.b>0.a+b=2.$$$$\mathrm{Find}\mathrm{the}\mathrm{minimum}\mathrm{value}\mathrm{of}{a}^{\sqrt{a}}{b}^{\sqrt{b}}.$$

Answered by Ghisom last updated on 28/Oct/24

$$b=2-a$$$$a^{\sqrt{a}}{(2-a)}^{\sqrt{2-a}};0<a<2$$$$\mathrm{since}\underset{x\to 0}{\lim }{x}^{\sqrt{x}}=1\mathrm{the}\mathrm{minimum}\mathrm{occurs}\mathrm{at}$$$$\mathrm{the}\mathrm{center}\mathrm{of}\mathrm{the}\mathrm{interval}\Rightarrow$$$$\min \left(a^{\sqrt{a}}{b}^{\sqrt{b}}\right)=1\mathrm{at}a=b=1$$

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