Math Problem Statement

Gambarkan grafik kurva y = x^4 - 4x^3 dan tentukan titik kritis, titik belok, titik minimum/maksimumnya.

Solution

np import matplotlib.pyplot as plt from sympy import symbols, diff, solve

Define the function

x = symbols('x') y = x4 - 4*x3

Calculate the derivative and critical points

y_prime = diff(y, x) critical_points = solve(y_prime, x)

Calculate the second derivative for inflection points

y_double_prime = diff(y_prime, x) inflection_points = solve(y_double_prime, x)

Evaluate the function for the critical points to find min/max

critical_values = [(point, y.subs(x, point)) for point in critical_points]

Define the function for plotting

def f(x): return x4 - 4*x3

Generate values for the plot

x_vals = np.linspace(-1, 5, 500) y_vals = f(x_vals)

Plot the function

plt.figure(figsize=(8, 6)) plt.plot(x_vals, y_vals, label='y = x^4 - 4x^3', color='blue') plt.axhline(0, color='black', linewidth=0.5, linestyle='--') plt.axvline(0, color='black', linewidth=0.5, linestyle='--') plt.scatter([float(p) for p in critical_points], [float(y.subs(x, p)) for p in critical_points], color='red', label='Critical Points') plt.scatter([float(p) for p in inflection_points], [float(y.subs(x, p)) for p in inflection_points], color='green', label='Inflection Points') plt.title('Graph of y = x^4 - 4x^3') plt.xlabel('x') plt.ylabel('y') plt.legend() plt.grid(True) plt.show()

critical_points, criticalGrafik fungsi y=x44x3y = x^4 - 4x^3 telah dibuat.

Titik Kritis

  • Titik kritis terjadi pada x=0x = 0 dan x=3x = 3.
  • Nilai fungsi di titik kritis:
    • (0,0)(0, 0)
    • (3,27)(3, -27)

Titik Belok

  • Titik belok terjadi pada x=0x = 0 dan x=2x = 2.

Titik Minimum/Maksimum

  • Pada x=0x = 0, titik adalah minimum lokal (0,0)(0, 0).
  • Pada x=3x = 3, titik adalah maksimum lokal (3,27)(3, -27).

Silakan tanyakan jika Anda ingin penjelasan lebih lanjut atau analisis mendalam.

Pertanyaan terkait:

  1. Bagaimana cara menentukan titik kritis menggunakan turunan pertama?
  2. Apa perbedaan antara titik belok dan titik kritis?
  3. Bagaimana cara menentukan apakah titik kritis adalah minimum atau maksimum?
  4. Apa interpretasi grafik untuk y=x44x3y = x^4 - 4x^3 di sekitar x=0x = 0?
  5. Bagaimana aplikasi konsep titik kritis dalam optimasi?

Tip:

Gunakan turunan kedua untuk mengonfirmasi jenis titik (minimum, maksimum,

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Math Problem Analysis

Mathematical Concepts

Calculus
Critical Points
Inflection Points
Maxima and Minima

Formulas

First derivative: y' = d/dx of the function
Second derivative: y'' = d²/dx² of the function
Critical points: Solve y' = 0
Inflection points: Solve y'' = 0

Theorems

First Derivative Test (to find critical points)
Second Derivative Test (to classify minima and maxima)

Suitable Grade Level

Grades 11-12