3次関数の接線と接点の関係について書いておきます。
3次関数の
における接線
は接点
と接点以外の共有点
を持つわけで, 2つのグラフの概形を描くと次のようになります。グラフ中の
は変曲点の
座標です。
念のため変曲点の座標の求めておくと,
として,
を得る。
このとき,
![Rendered by QuickLaTeX.com f(x)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d2fed79cedb38d253149db96430fffcb_l3.png)
![Rendered by QuickLaTeX.com y=mx+n](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-cfea608bc79c9784af52dba1465a0885_l3.png)
![Rendered by QuickLaTeX.com ax^3+bx^2+cx+d=mx+n](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-2cb7e76a3e7f963af85a4848bd7f44ec_l3.png)
![Rendered by QuickLaTeX.com ax^3+bx^2+cx+d-(mx+n)=0\cdots\maru1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5053f9ffb1ff937dc1512fe716724459_l3.png)
となるが, この左辺の式は, 接線と
![Rendered by QuickLaTeX.com f(x)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d2fed79cedb38d253149db96430fffcb_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com \alpha, \beta](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-a4b2ae318bdcaca10ccf19df480f6d8b_l3.png)
![Rendered by QuickLaTeX.com a(x-\alpha)^2(x-\beta)=0\cdots\maru2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-aa02e78c2f6c931ce0971652bf685f72_l3.png)
![Rendered by QuickLaTeX.com \maru1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c8b8ec9c0d15342374d474f3407d687d_l3.png)
![Rendered by QuickLaTeX.com \maru2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b55f3d993b1bbf2d3a2ca1e85ea19bd7_l3.png)
![Rendered by QuickLaTeX.com \maru1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c8b8ec9c0d15342374d474f3407d687d_l3.png)
![Rendered by QuickLaTeX.com ax^3+\underline{b}x^2+(c-m)x+d-n=0\cdots\maru1 '](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-acd802d673234a8c47a96b2016d4a76c_l3.png)
![Rendered by QuickLaTeX.com \maru2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b55f3d993b1bbf2d3a2ca1e85ea19bd7_l3.png)
![Rendered by QuickLaTeX.com a(x^2-2\alpha x+\alpha^2)(x-\beta)=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b2d84a7e9cf1f6ff92b9d2753487d630_l3.png)
![Rendered by QuickLaTeX.com a\left\{x^3+(-2\alpha-\beta)x^2+(2\alpha\beta+\alpha^2)x-\alpha^2\beta\right\}=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7e5d67245a2b936898e43ccd8b2267e1_l3.png)
![Rendered by QuickLaTeX.com ax^3+\underline{a(-2\alpha-\beta)}x^2+a(2\alpha\beta+\alpha^2)x-\alpha^2\beta a=0\cdots\maru2 '](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-ff840cd5f0f0b570ba8c089a654b732c_l3.png)
ここで,
![Rendered by QuickLaTeX.com \maru1 ', \maru2 '](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f605abd78b4652a80ba59332c311e0dd_l3.png)
![Rendered by QuickLaTeX.com x^2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-6a32b89cea4d53c2eda428b80831cfa8_l3.png)
![Rendered by QuickLaTeX.com b=a(-2\alpha-\beta)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-8e8b1cb4676332c8ab9b9057f9051443_l3.png)
![Rendered by QuickLaTeX.com \beta=-\dfrac{b}{a}-2\alpha\cdots\maru3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f1a3e55ac99415cbcbaa9a0c28244d6d_l3.png)
となる。これで接点の
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com m, n](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f1167b90c6de7ee47693f8b8e91efcee_l3.png)
また,
![Rendered by QuickLaTeX.com \maru3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-59e0023e63fc0665bd608c9e370731c9_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com \beta-\left(-\dfrac{b}{3a}\right)=2\left\{\left(-\dfrac{b}{3a}\right)-\alpha\right\}\cdots\maru3 '](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-e3792f506f73b353404ced9243d96d28_l3.png)
と変形でき, これの意味するところは,
![Rendered by QuickLaTeX.com \beta](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-28f9b40a3308d16b41465da4899b77d9_l3.png)
![Rendered by QuickLaTeX.com \alpha](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9da9c1b46f8817a14a1dc17dab7d2a84_l3.png)
これは, グラフの対称性や性質などから, 次のように表せる。
このように8個の合同な平行四辺形ができる。このような関係を知っておくと面積を求めるときの接点の座標の関係など便利なことが多い。
3次関数の接線と接点
3次関数のグラフ上の点
における接線が,
のグラフと再び交わる点を
とすると,
の関係が成り立つ。
これは, 接点から変曲点
までの水平方向の距離と変曲点
から接線の交点
までの水平距離の比が
であることを意味している。