数学 - Python - 微分積分学 - 平均値の定理 - 関数の値の変動、累乗、分数、累乗根、絶対値

学習環境

微分積分学 (ちくま学芸文庫) (吉田洋一(著)、筑摩書房)のⅢ.(平均値の定理)、演習問題Ⅲ、問6、7、8、9、10.の解答を求めてみる。

$\begin{array}{l} f (x) = 3 x^{4} - 4 x^{3} - 12 x^{2} + 5 \\ f' (x) = 12 x^{3} - 12 x^{2} - 24 x \\ = 12 x (x^{2} - x - 2) \\ = 12 x (x - 2) (x + 1) \\ f (- 1) = 3 + 4 - 12 + 5 = 0 \\ f (0) = 5 \\ f (2) = 48 - 32 - 48 + 5 = - 27 \end{array}$

よって極大値は5、極小値は 0、-27。
$\begin{array}{l} f (x) = \frac{x^{2} - x + 2}{x^{2} + x + 2} \\ f' (x) = \frac{(2 x - 1) (x^{2} + x + 2) - (x^{2} - x + 2) (2 x + 1)}{{(x^{2} + x + 2)}^{2}} \\ = \frac{(2 x^{3} + x^{2} + 3 x - 2) - (2 x^{3} - x^{2} + 3 x + 2)}{{(x^{2} + x + 2)}^{2}} \\ = \frac{2 x^{2} - 4}{{(x^{2} + x + 2)}^{2}} \\ = \frac{2 (x - \sqrt{2}) (x + \sqrt{2})}{{(x^{2} + x + 2)}^{2}} \\ f (- \sqrt{2}) = \frac{2 + \sqrt{2} + 2}{2 - \sqrt{2} + 2} \\ = \frac{4 + \sqrt{2}}{4 - \sqrt{2}} \\ = \frac{18 + 8 \sqrt{2}}{16 - 2} \\ = \frac{9 + 4 \sqrt{2}}{7} \\ f (\sqrt{2}) = \frac{9 - 4 \sqrt{2}}{7} \end{array}$

よって、極大値は

$\frac{9 + 4 \sqrt{2}}{7}$

極小値は

$\frac{9 - 4 \sqrt{2}}{7}$
$\begin{array}{l} f (x) = \sqrt[3]{(x - 1) {(x - 2)}^{2}} \\ f' (x) = \frac{1}{3} (x - 1) {(x - 2)}^{2} ({(x - 2)}^{2} + (x - 1) 2 (x - 2)) \\ = \frac{1}{3} (x - 1) {(x - 2)}^{3} (x - 2 + 2 (x - 1)) \\ = \frac{1}{3} (x - 1) {(x - 2)}^{3} (3 x - 4) \\ f (1) = 0 \\ f (\frac{4}{3}) = {(\frac{1}{3} {(\frac{2}{3})}^{2})}^{3} = \frac{4^{\frac{1}{3}}}{3} \\ f (2) = 0 \end{array}$

よって極小値は0、極大値は

$\frac{4^{\frac{1}{3}}}{3}$
$\begin{array}{l} f (x) = 2 x^{3} + 3 x^{2} - 12 x - 10 \\ f' (x) = 6 x^{2} + 6 x - 12 \\ = 6 (x^{2} + x - 2) \\ = 6 (x + 2) (x - 1) \\ f (- 2) = - 16 + 12 + 24 - 10 = 10 \\ f (1) = 2 + 3 - 12 - 10 = - 17 \end{array}$

よって、求める極大値は10、17。極小値は0。
$\begin{array}{l} f (x) = x^{3} - 3 x + 8 \\ f' (x) = 3 x^{2} - 3 \\ = 3 (x + 1) (x - 1) \\ f (- 1) = - 1 + 3 + 8 = 10 \\ f (1) = 1 - 3 + 8 = 6 \end{array}$

よって求める極大値は10。極小値は0、6。

#!/usr/bin/env python3 from sympy import pprint, symbols, plot, root, sqrt print('6.') x = symbols('x') fs = [3 * x ** 4 - 4 * x ** 3 - 12 * x ** 2 + 5, (x ** 2 - x + 2) / (x ** 2 + x + 2), root((x - 1) * (x - 2) ** 2, 3), abs(2 * x ** 3 + 3 * x ** 2 - 12 * x - 10), abs(x ** 3 - 3 * x + 8)] cs = [5, 0, -27, (9 + 4 * sqrt(2)) / 7, (9 - 4 * sqrt(2)) / 7, root(4, 3) / 3, 0, 10, 17, 0, 10, 0, 60, ] p = plot(*fs, 5, 0, -27, (9 + 4 * sqrt(2)) / 7, (9 - 4 * sqrt(2)) / 7, root(4, 3) / 3, 0, 10, 17, 0, 10, 0, 60, (x, -4, 4), ylim=(-30, 20), legend=False, show=False) colors = ['red', 'green', 'blue', 'brown', 'orange', 'purple', 'pink', 'gray', 'skyblue', 'yellow'] for o, color in zip(p, colors): o.line_color = color for o, color in zip(fs + cs, colors): print(o, color) p.show() p.save('sample6.png')

入出力結果(Zsh、cmd.exe(コマンドプロンプト)、Terminal、Jupyter(IPython))

% ./sample6.py 6. 3*x**4 - 4*x**3 - 12*x**2 + 5 red (x**2 - x + 2)/(x**2 + x + 2) green ((x - 2)**2*(x - 1))**(1/3) blue Abs(2*x**3 + 3*x**2 - 12*x - 10) brown Abs(x**3 - 3*x + 8) orange 5 purple 0 pink -27 gray 4*sqrt(2)/7 + 9/7 skyblue 9/7 - 4*sqrt(2)/7 yellow %

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2019年11月28日木曜日

数学 - Python - 微分積分学 - 平均値の定理 - 関数の値の変動、累乗、分数、累乗根、絶対値

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