2017年2月17日金曜日

数学 - 解析学 - 微分法 - 平均値の定理(放物線、法線、線分の長さ)

$\begin{array}{l}A\left(a,{a}^{2}\right),A\left(b,{b}^{2}\right)\\ y\text{'}=2x\\ y=-\frac{1}{2a}x+c\\ {a}^{2}=-\frac{1}{2a}a+c\\ c={a}^{2}+\frac{1}{2a}\\ y=-\frac{1}{2a}x+{a}^{2}+\frac{1}{2a}\\ {x}^{2}=-\frac{1}{2a}x+{a}^{2}+\frac{1}{2a}\\ {x}^{2}+\frac{1}{2a}x-{a}^{2}+\frac{1}{2a}=0\\ ab=-{a}^{2}+\frac{1}{2a}\\ a+b=-\frac{1}{2a}\\ b=-a-\frac{1}{2a}=-\frac{2{a}^{2}+1}{2a}\\ {b}^{2}={\left(\frac{2{a}^{2}+1}{2a}\right)}^{2}\\ f\left(a\right)={\left(a-b\right)}^{2}+{\left({a}^{2}-{b}^{2}\right)}^{2}\\ ={\left(a-b\right)}^{2}\left(1+{\left(a+b\right)}^{2}\right)\\ =\left({a}^{2}-2ab+{b}^{2}\right)\left(1+{a}^{2}+2ab+{b}^{2}\right)\\ =\left({a}^{2}+2{a}^{2}+{\left(\frac{2{a}^{2}+1}{2a}\right)}^{2}\right)\left(1+{a}^{2}-2{a}^{2}-1+{\left(\frac{2{a}^{2}+1}{2a}\right)}^{2}\right)\\ =\left(3{a}^{2}+{\left(\frac{2{a}^{2}+1}{2a}\right)}^{2}\right)\left(-{a}^{2}+{\left(\frac{2{a}^{2}+1}{2a}\right)}^{2}\right)\\ f\text{'}\left(a\right)=\left(6a+\frac{2{a}^{2}+1}{2a}·\frac{2{a}^{2}-1}{{a}^{2}}\right)\left(-2a+\frac{2{a}^{2}+1}{2a}·\frac{2{a}^{2}-1}{{a}^{2}}\right)\\ =\left(6a+\frac{4{a}^{2}-1}{2{a}^{3}}\right)\left(-2a+\frac{4{a}^{2}-1}{2{a}^{3}}\right)\\ =\frac{12{a}^{4}+4{a}^{2}-1}{2{a}^{3}}·\frac{-4{a}^{4}+4{a}^{2}-1}{2{a}^{3}}\\ 12{a}^{4}+4{a}^{2}-1=0\\ \left(6{a}^{2}-1\right)\left(2{a}^{2}+1\right)=0\\ {a}^{2}=\frac{1}{6}\\ 4{a}^{4}-4{a}^{2}+1=0\\ {\left(2{a}^{2}-1\right)}^{2}=0\\ {a}^{2}=\frac{1}{2}\\ a=±\frac{1}{\sqrt{2}},±\frac{1}{\sqrt{6}}\\ \\ A\left(±\frac{1}{\sqrt{2}},\frac{1}{2}\right)\\ b=-\frac{2{a}^{2}+1}{2a}=\mp \frac{2}{\frac{2}{\sqrt{2}}}=\mp \sqrt{2}\\ {b}^{2}=2\\ \sqrt{{\left(\frac{1}{\sqrt{2}}+\sqrt{2}\right)}^{2}+{\left(2-\frac{1}{2}\right)}^{2}}\\ =\sqrt{\frac{1}{2}+2+2+4+\frac{1}{4}-2}\\ =\frac{3\sqrt{3}}{2}\end{array}$