主要内容
拐点复习
回顾一下你对拐点的认识, 以及我们如何使用微分方法来找到它们。
练习2:以代数方法来分析拐点
发现拐点的方式类似于我们找到极值点。然而,我们不是在寻找改变导数符号的点,而是在寻找改变二阶导数符号的点。
例如,让我们找到f, left parenthesis, x, right parenthesis, equals, start fraction, 1, divided by, 2, end fraction, x, start superscript, 4, end superscript, plus, x, cubed, minus, 6, x, squared的拐点。
f的二阶导数是f, start superscript, prime, prime, end superscript, left parenthesis, x, right parenthesis, equals, 6, left parenthesis, x, minus, 1, right parenthesis, left parenthesis, x, plus, 2, right parenthesis。
当x, equals, minus, 2, comma, 1,f, start superscript, prime, prime, end superscript, left parenthesis, x, right parenthesis, equals, 0,且它在任何位置都有定义。 x, equals, minus, 2和x, equals, 1将数轴划分为三个区间:
让我们评估每区间的f, start superscript, prime, prime, end superscript,以查看它在该区间是正数还是负数。
区间 | x-值 | f, start superscript, prime, prime, end superscript, left parenthesis, x, right parenthesis | 判定 |
---|---|---|---|
x, is less than, minus, 2 | x, equals, minus, 3 | f, start superscript, prime, prime, end superscript, left parenthesis, minus, 3, right parenthesis, equals, 24, is greater than, 0 | f上凹 \cup |
minus, 2, is less than, x, is less than, 1 | x, equals, 0 | f, start superscript, prime, prime, end superscript, left parenthesis, 0, right parenthesis, equals, minus, 12, is less than, 0 | f下凹 \cap |
x, is greater than, 1 | x, equals, 2 | f, start superscript, prime, prime, end superscript, left parenthesis, 2, right parenthesis, equals, 24, is greater than, 0 | f上凹 \cup |
我们可以看到f在x, equals, minus, 2和x, equals, 1改变了图像凹凸性,所以f的拐点就是这两个x-值。
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