a-gt-0-b-gt-0-a-b-2-Find-the-minimum-value-of-a-a-b-b-

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

a > 0 . b > 0 . a + b = 2 .

Find the minimum value 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

since \lim_{x \to 0} x^{\sqrt{x}} = 1 the minimum occurs at

the center of the interval \Rightarrow

\min (a^{\sqrt{a}} b^{\sqrt{b}}) = 1 at a = b = 1

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