数学 - Python - 代数学 - 1次関数、2次関数 - 2次関数の最大と最小 - 頂点

学習環境

代数への出発 (新装版数学入門シリーズ) (松坂和夫(著)、岩波書店)の第6章(1次方程式、2次方程式)、5(2次関数の最大と最小)の問14の解答を求めてみる。

1. $\begin{array}{l} y = x^{2} - 4 x + 7 \\ = {(x - 2)}^{2} + 3 \end{array}$
  
  よって
  
  $x = 2$
  
  のとき最小となり、その値は3。
2. $y = - {(x + 3)}^{2} + 14$
  
  x が-3のとき最大値14。
3. $y = 4 {(x + \frac{3}{2})}^{2}$
  
  よって
  
  $x = - \frac{3}{2}$
  
  のとき最小値0。
4. $\begin{array}{l} y = \frac{3}{4} (x^{2} + 8 x) - 12 \\ = \frac{3}{4} {(x + 4)}^{2} - 24 \end{array}$
  
  x が-4のとき最小値-24。
5. $\begin{array}{l} y = - \frac{3}{2} (x^{2} - \frac{4}{3} x) \\ = - \frac{3}{2} {(x - \frac{2}{3})}^{2} + \frac{3}{2} {(\frac{2}{3})}^{2} \\ = - \frac{3}{2} {(x - \frac{2}{3})}^{2} + \frac{2}{3} \end{array}$
6. $\begin{array}{l} y = \frac{1}{3} (x - \frac{3}{2}) (- \frac{1}{2}) (x - \frac{2}{5}) \\ = - \frac{1}{6} (x^{2} - (\frac{3}{2} + \frac{2}{5}) x) - \frac{1}{10} \\ = - \frac{1}{6} (x^{2} - \frac{19}{10} x) - \frac{1}{10} \\ = - \frac{1}{6} {(x - \frac{19}{20})}^{2} + \frac{1}{6} \cdot {(\frac{19}{20})}^{2} - \frac{1}{10} \\ = - \frac{1}{6} {(x - \frac{19}{20})}^{2} + \frac{1}{6} \cdot \frac{361}{400} - \frac{1}{10} \\ = - \frac{1}{6} {(x - \frac{19}{20})}^{2} + \frac{121}{2400} \end{array}$
7. $\begin{array}{l} y = a (x^{2} - \frac{b}{a} x) \\ = a {(x - \frac{b}{2 a})}^{2} - \frac{b^{2}}{4 a} \end{array}$
  
  よって
  
  $x = - \frac{b}{2 a}$
  
  のとき最小値
  
  $- \frac{b^{2}}{4 a}$
8. $\begin{array}{l} y = a (x^{2} - x) \\ = a {(x - \frac{1}{2})}^{2} - \frac{a}{4} \end{array}$

#!/usr/bin/env python3 from sympy import symbols, plot, Rational print('14.') x = symbols('x') ys = [x ** 2 - 4 * x + 7, 4 * x ** 2 + 12 * x + 9, x * (4 - 3 * x) / 2] ys0 = [3, 0, Rational(2, 3)] p = plot(*ys, *ys0, (x, -10, 10), ylim=(-10, 10), legend=True, show=False) colors = ['red', 'green', 'blue', 'brown', 'orange', 'purple', 'pink', 'gray', 'skyblue', 'yellow'] for o, color in zip(p, colors): o.line_color = color p.show() p.save('sample14.png')

% ./sample14.py 14. /opt/local/Library/Frameworks/Python.framework/Versions/3.8/lib/python3.8/site-packages/sympy/plotting/plot.py:1065: MatplotlibDeprecationWarning: The set_smart_bounds function was deprecated in Matplotlib 3.2 and will be removed two minor releases later. self.ax[i].spines['left'].set_smart_bounds(True) /opt/local/Library/Frameworks/Python.framework/Versions/3.8/lib/python3.8/site-packages/sympy/plotting/plot.py:1066: MatplotlibDeprecationWarning: The set_smart_bounds function was deprecated in Matplotlib 3.2 and will be removed two minor releases later. self.ax[i].spines['bottom'].set_smart_bounds(False) %

Kamimura's blog

2020年6月1日月曜日

数学 - Python - 代数学 - 1次関数、2次関数 - 2次関数の最大と最小 - 頂点

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