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Question Number 171529 by cortano1 last updated on 17/Jun/22

Commented by kapoorshah last updated on 17/Jun/22

$$25$$

Answered by FelipeLz last updated on 17/Jun/22

$$4y+9x=xy$$$$x=\frac{4y}{y-9}\{x>0\iff y>9\}$$$$s=x+y=\frac{4y}{y-9}+y=\frac{y^2-5y}{y-9}$$$$\frac{ds}{dy}=\frac{y^2-18y+45}{{(y-9)}^2}\{\frac{ds}{dy}=0\iff y^2-18y+45=0\}$$$$y=\{\begin{array}{l}15\\ \cancel{3}\end{array}$$$$s=25$$

Commented by kapoorshah last updated on 17/Jun/22

$$nice$$

Answered by som(math1967) last updated on 17/Jun/22

$$\frac{4}{x}+\frac{9}{y}=1$$$$y=\frac{9x}{x-4}$$$$x+y$$$$=x+\frac{9x}{x-4}=f\left(x\right)let$$$$f^l\left(x\right)=1+\frac{9(x-4)-9x}{{(x-4)}^2}=1-\frac{36}{{(x-4)}^2}$$$$f^{\mathit{l}\mathit{l}}\left(\mathit{x}\right)=\frac{72}{{(\mathit{x}-4)}^3}$$$$\mathit{f}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{x}\mathit{o}\mathit{r}\mathit{m}\mathit{i}\mathit{n}{\mathit{f}}^{\mathit{l}}\left(\mathit{x}\right)=0$$$$\therefore 1-\frac{36}{{(\mathit{x}-4)}^2}=0$$$$\Rightarrow (x-4)=\pm 6\therefore x=10or-2$$$${\mathit{f}}^{\mathit{l}\mathit{l}}\left(10\right)=\frac{72}{6^3}>0$$$$\therefore min(x+y)=10+\frac{90}{6}=25$$$$$$

Commented by kapoorshah last updated on 17/Jun/22

$$nice$$

Answered by greougoury555 last updated on 18/Jun/22

$$\frac{4}{x}+\frac{9}{y}=1$$$$\frac{2^2}{x}+\frac{3^2}{y}=1$$$$1⩾\frac{{(2+3)}^2}{x+y}$$$$\Rightarrow x+y⩾25\left(min\right)$$$$when\frac{2}{x}=\frac{3}{y}=\frac{1}{5}$$

Answered by floor(10²Eta[1]) last updated on 17/Jun/22

$$4\mathrm{y}+9\mathrm{x}=\mathrm{xy}$$$$\mathrm{AM}⩾\mathrm{GM}:$$$$\frac{4\mathrm{y}+9\mathrm{x}}{2}⩾\sqrt{\left(4\mathrm{y}\right)\left(9\mathrm{x}\right)}$$$$\frac{\mathrm{xy}}{2}⩾\sqrt{36\mathrm{xy}}$$$$\mathrm{xy}⩾12\sqrt{\mathrm{xy}}$$$${\left(\mathrm{xy}\right)}^2⩾144\mathrm{xy}$$$$\mathrm{xy}(\mathrm{xy}-144)⩾0\Rightarrow \mathrm{xy}⩾144$$$$$$$$\frac{\mathrm{x}+\mathrm{y}}{2}⩾\sqrt{\mathrm{xy}}⩾\sqrt{144}=12$$$$\mathrm{x}+\mathrm{y}⩾24$$$$$$

Commented by mr W last updated on 17/Jun/22

$$i{don}^{\prime }tthinkthisiscorrect.$$$$in\frac{4\mathrm{y}+9\mathrm{x}}{2}⩾\sqrt{\left(4\mathrm{y}\right)\left(9\mathrm{x}\right)}$$$$‘‘{=}^{″}isvalidwhen4y=9x,i.e.x=\frac{4}{9}y.$$$$butin\frac{\mathrm{x}+\mathrm{y}}{2}⩾\sqrt{\mathrm{xy}}$$$$‘‘{=}^{″}isvalidwhenx=y.$$

Commented by cortano1 last updated on 18/Jun/22

$$thecorrectansweris25$$

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