if-a-b-c-gt-0-and-a-b-c-1-find-min-1-a-9-b-25-c-a-73-b-75-c-105-d-81-e-83-

Question Number 150880 by mathdanisur last updated on 16/Aug/21

if a; b; c > 0 and a + b + c = 1

find \min (\frac{1}{a} + \frac{9}{b} + \frac{25}{c}) = ?

a) 7 3 b) 75 c) 105 d) 81 e) 83

Answered by mr W last updated on 16/Aug/21

s = \frac{1}{a} + \frac{9}{b} + \frac{25}{1 - a - b}

\frac{\partial s}{\partial a} = - \frac{1}{a^{2}} + \frac{25}{{(1 - a - b)}^{2}} = 0

\frac{1}{a} = \frac{5}{1 - a - b}

\frac{\partial s}{\partial b} = - \frac{9}{b^{2}} + \frac{25}{{(1 - a - b)}^{2}} = 0

\frac{3}{b} = \frac{5}{1 - a - b} = \frac{1}{a}

\Rightarrow b = 3 a

\Rightarrow a = \frac{1}{9}

\Rightarrow b = \frac{1}{3}

\Rightarrow c = \frac{5}{9}

s_{m i n} = 9 + 9 \times 3 + 25 \times \frac{9}{5} = 81

Commented by mathdanisur last updated on 16/Aug/21

Thank you Ser

Answered by qaz last updated on 16/Aug/21

By Cauthy Schwarz inequality in engel form :

\frac{1}{a} + \frac{9}{b} + \frac{25}{c} ⩾ \frac{{(1 + 3 + 5)}^{2}}{a + b + c} = 81

Commented by mathdanisur last updated on 16/Aug/21

Thank you Ser