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

$$\mathrm{if}\mathrm{a};\mathrm{b};\mathrm{c}>0\mathrm{and}\mathrm{a}+\mathrm{b}+\mathrm{c}=1$$$$\mathrm{find}\mathbf{min}(\frac{1}{\mathrm{a}}+\frac{9}{\mathrm{b}}+\frac{25}{\mathrm{c}})=?$$$$\mathrm{a})73\mathrm{b})75\mathrm{c})105\mathrm{d})81\mathrm{e})83$$

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

$$s=\frac{1}{a}+\frac{9}{b}+\frac{25}{1-a-b}$$$$\frac{\partial s}{\partial a}=-\frac{1}{a^2}+\frac{25}{{(1-a-b)}^2}=0$$$$\frac{1}{a}=\frac{5}{1-a-b}$$$$\frac{\partial s}{\partial b}=-\frac{9}{b^2}+\frac{25}{{(1-a-b)}^2}=0$$$$\frac{3}{b}=\frac{5}{1-a-b}=\frac{1}{a}$$$$\Rightarrow b=3a$$$$\Rightarrow a=\frac{1}{9}$$$$\Rightarrow b=\frac{1}{3}$$$$\Rightarrow c=\frac{5}{9}$$$$s_{min}=9+9\times 3+25\times \frac{9}{5}=81$$

Commented by mathdanisur last updated on 16/Aug/21

$$\mathrm{Thank}\mathrm{you}\mathrm{Ser}$$

Answered by qaz last updated on 16/Aug/21

$$\mathrm{By}\mathrm{Cauthy}\mathrm{Schwarz}\mathrm{inequality}\mathrm{in}\mathrm{engel}\mathrm{form}:$$$$\frac{1}{\mathrm{a}}+\frac{9}{\mathrm{b}}+\frac{25}{\mathrm{c}}⩾\frac{{(1+3+5)}^2}{\mathrm{a}+\mathrm{b}+\mathrm{c}}=81$$

Commented by mathdanisur last updated on 16/Aug/21

$$\mathrm{Thank}\mathrm{you}\mathrm{Ser}$$

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