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[不等式] 与方程有解相关的最值问题

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力工 posted 2024-10-3 18:29 |Read mode
已知$a,b,c>0$且关于$x$的方程$ax^2+bx+c=0$有解,求$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$的最小值。

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kuing posted 2024-10-3 22:41
题目最好说明是实数解,毕竟虚数解也是解。

将 `a/b+b/c+c/a` 看成 `b` 的函数 `f(b)`,显然 `f(b)` 在 `[\sqrt{ac},+\infty)` 递增,而由条件有 `b\geqslant2\sqrt{ac}`,所以
\[f(b)\geqslant f(2\sqrt{ac})=\frac a{2\sqrt{ac}}+\frac{2\sqrt{ac}}c+\frac ca
=\frac54\sqrt{\frac ac}+\frac54\sqrt{\frac ac}+\frac ca\geqslant3\sqrt[3]{\frac{25}{16}}.\]

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还得是k神出手,我一直只想一个个消$b$,想不到整体消。  posted 2024-10-4 07:52

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敬畏数学 posted 2024-10-4 12:56
不难!很简单。
还得是k神出手,我一直只想一个个消,想不到整体消。是这样啊

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