The minimum value of ax + by where xy = c^2 i | vedclass

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(A) Let $f(x, y) = ax + by$ subject to the constraint $xy = c^2$,where $x, y > 0$.Substituting $y = \frac{c^2}{x}$ into the expression,we get $f(x) =

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$2c\sqrt{ab}$

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(A) Let $f(x, y) = ax + by$ subject to the constraint $xy = c^2$,where $x, y > 0$.Substituting $y = \frac{c^2}{x}$ into the expression,we get $f(x) = ax + \frac{bc^2}{x}$.To find the minimum,we differentiate with respect to $x$:$f'(x) = a - \frac{bc^2}{x^2}$.Setting $f'(x) = 0$,we get $a = \frac{bc^2}{x^2}$,which implies $x^2 = \frac{bc^2}{a}$,so $x = c\sqrt{\frac{b}{a}}$.Since $x, y > 0$,we take the positive root.Then $y = \frac{c^2}{x} = \frac{c^2}{c\sqrt{b/a}} = c\sqrt{\frac{a}{b}}$.The minimum value is $f = a(c\sqrt{\frac{b}{a}}) + b(c\sqrt{\frac{a}{b}}) = c\sqrt{ab} + c\sqrt{ab} = 2c\sqrt{ab}$.

The minimum value of $ax + by$ where $xy = c^2$ is

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