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Question Number 44898 by ajfour last updated on 06/Oct/18

$$If{x}^4+{px}^3+{qx}^2+rx+5=0$$$$hasfourrealroots,thenfind$$$$theminimumvalueof\mathit{p}\mathit{r}.$$

Commented by MJS last updated on 06/Oct/18

$$\mathrm{let}\mathrm{me}\mathrm{try}...$$

Answered by ajfour last updated on 06/Oct/18

$$x_1{x}_2{x}_3{x}_4=5$$$$\mathrm{\Sigma }{x}_1{x}_2{x}_3=-r$$$$x_1+x_2+x_3+x_4=-p$$$$pr=5(\frac{1}{x_4}+\frac{1}{x_2}+\frac{1}{x_3}+\frac{1}{x_4})\mathrm{\Sigma }{x}_1$$$$>5\times \frac{4}{x_c}\times 4x_c(=80).$$$$Hencemin\left(\mathit{p}\mathit{r}\right)=80.$$$$$$

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