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Question Number 143234 by mathdanisur last updated on 11/Jun/21

Answered by mr W last updated on 11/Jun/21

$$F=x^3+y^3+z^3+\lambda (\alpha x+\alpha y+\gamma z-1)$$$$\frac{\partial F}{\partial x}=3x^2+\lambda \alpha =0$$$$\Rightarrow x=\sqrt{-\frac{\lambda \alpha }{3}}$$$$similarly$$$$\Rightarrow y=\sqrt{-\frac{\lambda \beta }{3}}$$$$\Rightarrow z=\sqrt{-\frac{\lambda \gamma }{3}}$$$$\Rightarrow (\alpha \sqrt{\alpha }+\beta \sqrt{\beta }+\gamma \sqrt{\gamma })\sqrt{-\frac{\lambda }{3}}=1$$$$\Rightarrow \sqrt{-\frac{\lambda }{3}}=\frac{1}{\alpha \sqrt{\alpha }+\beta \sqrt{\beta }+\gamma \sqrt{\gamma }}$$$$\Rightarrow x=\frac{\sqrt{\alpha }}{\alpha \sqrt{\alpha }+\beta \sqrt{\beta }+\gamma \sqrt{\gamma }}$$$$\Rightarrow y=\frac{\sqrt{\beta }}{\alpha \sqrt{\alpha }+\beta \sqrt{\beta }+\gamma \sqrt{\gamma }}$$$$\Rightarrow z=\frac{\sqrt{\gamma }}{\alpha \sqrt{\alpha }+\beta \sqrt{\beta }+\gamma \sqrt{\gamma }}$$$$P_{min}=\frac{\alpha \sqrt{\alpha }+\beta \sqrt{\beta }+\gamma \sqrt{\gamma }}{{(\alpha \sqrt{\alpha }+\beta \sqrt{\beta }+\gamma \sqrt{\gamma })}^3}=\frac{1}{{(\alpha \sqrt{\alpha }+\beta \sqrt{\beta }+\gamma \sqrt{\gamma })}^2}$$

Commented by mathdanisur last updated on 12/Jun/21

$$thaksSir,sorrypleasecheckagain..$$

Commented by mr W last updated on 12/Jun/21

$$whatshallicheck?ismyanswer$$$$wrong?$$

Commented by mathdanisur last updated on 12/Jun/21

$$yesSir,pleasesolveagain..$$

Commented by mr W last updated on 12/Jun/21

$$i{can}^{\prime }tfindmistakeinmyanswer.$$$$if{it}^{\prime }swrong,pleasetellmetheright$$$$answer.$$

Commented by mathdanisur last updated on 13/Jun/21

$$thankyousomuchdearSir..$$

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