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[函数] 请教一个最大值的最小值

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hongxian posted 2013-8-14 15:21 |Read mode
函数$F(x)=\abs{\cos^2x+2\sin x\cos x-\sin^2x+ax+b}$在$0\leqslant x\leqslant\dfrac32\pi$上的最大值为$M(a,b)$,求$M(a,b)$的最小值,并求此时$a$、$b$的值。

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爪机专用 posted 2013-8-14 19:40
好像旧论坛有类似,等会儿电脑上再找找

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kuing posted 2013-8-14 20:17
好像旧论坛有类似,等会儿电脑上再找找
爪机专用 发表于 2013-8-14 19:40
就是这个 kkkkuingggg.haotui.com/viewthread.php?tid=1400 不知有多大关系,看看先。

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kuing posted 2013-8-14 20:57
噢,原来现在这个比较简单,想象一下图像,然后常规地先猜后证。

\begin{align*}
4M(a,b)&\geqslant F\left(\frac\pi8\right)+2F\left(\frac{5\pi}8\right)+F\left( \frac{9\pi}8\right) \\
& =\left|\sqrt2+\frac{\pi a}8+b\right|+2\left|-\sqrt2+\frac{5\pi a}8+b \right|+\left|\sqrt2+\frac{9\pi a}8+b\right| \\
& \geqslant \left|\sqrt2+\frac{\pi a}8+b-2\left(-\sqrt2+\frac{5\pi a}8+b \right)+\sqrt2+\frac{9\pi a}8+b\right| \\
& =4\sqrt2,
\end{align*}
即 $M(a,b)\geqslant \sqrt2$,当 $a=b=0$ 时取得最小值。

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其妙 posted 2013-8-17 13:33
就是这个  不知有多大关系,看看先。
kuing 发表于 2013-8-14 20:17
旧版好多广告哦!

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kuing posted 2013-8-17 14:21
回复 5# 其妙

假如回那边发展,你能忍受不?

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其妙 posted 2013-8-19 16:47
回复 6# kuing
勉强可以忍受,但是两相比较,还是这边安逸些

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其妙 posted 2013-8-19 16:50
这是1983年全国MO

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