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解决这个数学问题

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سؤال

image.pngimage.png

解决这个数学问题

Python

python
from PIL import Image
import matplotlib.pyplot as plt

img = Image.open('/mnt/data/image.png')
plt.imshow(img)
plt.axis('off')

(-0.5, 783.5, 139.5, -0.5)

Python

python
plt.figure(figsize=(3,2))
plt.imshow(img.crop((300,0,500,200)))
plt.axis('off')

(-0.5, 199.5, 199.5, -0.5)

الفكرة

Verifying calculation expression

The user has provided an image with Chinese text, asking to solve a math problem. It looks like the expression in the image is y = (x - 5) * x^{2/3}. I need to zoom in on the exponent to make sure it's actually 2/3. Using the image, it seems the exponent is written as 2 over 3, confirming it's 2/3 and the expression is accurate. Now I can move forward solving the problem.

Analyzing inflection point

The user wants to find the inflection point of the curve y = (x - 5) * x^(2/3). To do that correctly, we need to first compute the second derivative and solve for where concavity changes.

The key challenge is that the derivative of x^(2/3) involves negative exponents and may not be defined at x = 0. Since x^(2/3) is defined for all real x, we need to treat values where x is positive and negative separately for the derivative. I'll compute the derivatives now for both cases.

Finding inflection point at x=-1

We are looking for the inflection point of y = (x - 5) * |x|^{2/3}. After analyzing the second derivative, we see that it changes sign at x = -1, going from negative (concave down) to positive (concave up). In fact, at x = -1, the second derivative is defined, and the first derivative is also continuous. Thus, the inflection point is at x = -1, where the concavity changes.

Thought for 2m 14s

الإجابة

解题思路

设 y=(x−5) x23y=(x-5)\,x^{\tfrac23}，先求二阶导数，再判断凹凸性的变化。

1 . 一阶导数

y′=(x−5) ddx ⁣(x23)+x23 ddx(x−5)=(x−5)  23 x−13+x23  =  5x−103 x13  =  53(x−2) x−13.\begin{aligned} y' &= (x-5)\,\frac{d}{dx}\!\left(x^{\tfrac23}\right)+x^{\tfrac23}\,\frac{d}{dx}(x-5) \\[4pt] &= (x-5)\;\frac23\,x^{-\tfrac13}+x^{\tfrac23} \;=\;\frac{5x-10}{3\,x^{\tfrac13}} \;=\;\frac{5}{3}(x-2)\,x^{-\tfrac13}. \end{aligned}

除 x=0x=0 外，一阶导数在整个定义域内存在；x=2x=2 是驻点，但与拐点无关（见下文）。

2 . 二阶导数

y′′=ddx ⁣[53(x−2) x−13]=53[x−13−(x−2) 13 x−43]=109(x+1) x−43(x≠0).\begin{aligned} y'' &= \frac{d}{dx}\!\left[\frac{5}{3}(x-2)\,x^{-\tfrac13}\right] =\frac{5}{3}\Bigl[x^{-\tfrac13}-(x-2)\,\frac13\,x^{-\tfrac43}\Bigr] \\[4pt] &=\frac{10}{9}(x+1)\,x^{-\tfrac43}\qquad(x\ne0). \end{aligned}

同理，在 x<0x<0 时用 ∣x∣2/3|x|^{2/3} 的求导公式可得

y′′=109(x+1)(−x)−43,y''=\frac{10}{9}(x+1)(-x)^{-\tfrac43},

符号完全由 (x+1)(x+1) 决定。

3 . 凹凸性变化

区间y′′y'' 符号凹/凸
x<−1x<-1x+1<0⇒y′′<0x+1<0\Rightarrow y''<0凹向下
−1<x<0-1<x<0x+1>0⇒y′′>0x+1>0\Rightarrow y''>0凹向上
x>0x>0x+1>0⇒y′′>0x+1>0\Rightarrow y''>0凹向上
  • 在 x=−1x=-1 处，二阶导数 变号（由负变正），且一阶导数、函数值都有限，故此点是拐点。
  • x=0x=0 虽二阶导数不存在，但左右凹凸同为凹向上，故 不是 拐点。
  • x=2x=2 只是驻点（极小点），二阶导数大于零，无凹凸变化。

4 . 拐点坐标

x=−1,y=( −1−5) ∣−1∣23=(−6)⋅1=−6.\begin{aligned} x &= -1, \\ y &= (\,{-1}-5)\,|{-1}|^{\tfrac23}=(-6)\cdot1=-6. \end{aligned}
 (−1, −6) \boxed{\,(-1,\,-6)\,}

因此，曲线 y=(x−5)x23y=(x-5)x^{\tfrac23} 的拐点坐标是 (−1, −6)(-1,\,-6)。

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