## 2020年7月12日日曜日

### 数学 - Python - 代数学 - 不等式 - 不等式の証明 - 4次式、絶対不等式、非負、平方、式の変形、判別式

1. $\left({a}^{4}+{b}^{4}+{c}^{4}\right)-\left({b}^{2}{c}^{2}+{c}^{2}{a}^{2}+{a}^{2}{b}^{2}\right)$
$=\frac{1}{2}\left({\left({a}^{2}-{b}^{2}\right)}^{2}+{\left({b}^{2}-{c}^{2}\right)}^{2}+{\left({c}^{2}-{a}^{2}\right)}^{2}\right)$
$\ge 0$

よって、

${a}^{4}+{b}^{4}+{c}^{4}\ge {b}^{2}{c}^{2}+{c}^{2}{a}^{2}+{a}^{2}{b}^{2}$
$\left({b}^{2}{c}^{2}+{c}^{2}{a}^{2}+{a}^{2}{b}^{2}\right)-abc\left(a+b+c\right)$
$=\left({b}^{2}+{c}^{2}-bc\right){a}^{2}-bc\left(b+c\right)a+{b}^{2}{c}^{2}$
$D={\left(bc\left(b+c\right)\right)}^{2}-4\left({b}^{2}+{c}^{2}-bc\right){b}^{2}{c}^{2}$
$={b}^{2}{c}^{2}\left({\left(b+c\right)}^{2}-4\left({b}^{2}+{c}^{2}-bc\right)\right)$
$={b}^{2}{c}^{2}\left(-3{b}^{2}-3{c}^{2}+6bc\right)$
$=-3{b}^{2}{c}^{2}\left({b}^{2}+{c}^{2}-2bc\right)$
$=-3{b}^{2}{c}^{2}{\left(b-c\right)}^{2}$
$\le 0$

よって、

$\left({b}^{2}{c}^{2}+{c}^{2}{a}^{2}+{a}^{2}{b}^{2}\right)-abc\left(a+b+c\right)\ge 0$

ゆえに、

$abc\left(a+b+c\right)\le {b}^{2}{c}^{2}+{c}^{2}{a}^{2}+{a}^{2}{b}^{2}$

以上より、

${a}^{4}+{b}^{4}+{c}^{4}\ge {b}^{2}{c}^{2}+{c}^{2}{a}^{2}+{a}^{2}{b}^{2}\ge abc\left(a+b+c\right)$

（証明終）

コード

#!/usr/bin/env python3
from sympy.plotting import plot3d
from sympy.abc import x, y

print('19.')

z = 1
f = x ** 4 + y ** 4 + z ** 4
g = (y * z) ** 2 + (z * x) ** 2 + (x * y) ** 2
h = x * y * z * (x + y + z)
colors = ['red', 'green', 'blue', 'brown', 'orange',
'purple', 'pink', 'gray', 'skyblue', 'yellow']
p = plot3d(f, g, h,
(x, -5, 5),
(y, -5, 5),
show=False)
p.xlabel = x
p.ylabel = y
p.save(f'sample19.png')
p.show()


% ./sample19.py
19.
%