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 telah dibuat.
Titik Kritis
- Titik kritis terjadi pada dan .
- Nilai fungsi di titik kritis:
Titik Belok
- Titik belok terjadi pada dan .
Titik Minimum/Maksimum
- Pada , titik adalah minimum lokal .
- Pada , titik adalah maksimum lokal .
Silakan tanyakan jika Anda ingin penjelasan lebih lanjut atau analisis mendalam.
Pertanyaan terkait:
- Bagaimana cara menentukan titik kritis menggunakan turunan pertama?
- Apa perbedaan antara titik belok dan titik kritis?
- Bagaimana cara menentukan apakah titik kritis adalah minimum atau maksimum?
- Apa interpretasi grafik untuk di sekitar ?
- 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
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