Question

Исследовать функцию и построить график y=4x^3-2x

Answer 1

Исследование функции
Y = 4x³ - 2x
1. Область определения - Х⊂ R - разрывов (исключений) нет.
2. Корни функции   Y=2x*(2x² - 2) =0 - точки пересечения с осью Х.
x 1= 0  x2 = - 1/2*√2   x3 = 1/2*√2
3. Экстремумы функции - первая производная
Y' = 12x² - 2 = 0     x1 = - √1/6 ~ -0.41    x2 = √1/6 ~ 0.41
Максимум Ymax = Y(-0.41) = 0.544
Минимум Ymin = Y(0.41) = - 0.544
4. Точка перегиба - вторая производная.
Y" = 24*x = 0
При х<0 - выпуклая и х>0 - вогнутая
5. Область значений Y ⊂(-∞; +∞)

Answer 2

Function Exploration and Graph Plotting: y = 4x^3 - 2x

To explore the function y = 4x^3 - 2x and plot its graph, we can follow these steps:

1. Function Analysis: - The given function is y = 4x^3 - 2x. - We can analyze the function and its properties to understand its behavior and characteristics.

2. Graph Plotting: - After analyzing the function, we can plot its graph to visualize its behavior and shape.

Function Analysis:

The given function is a polynomial function of the form y = ax^3 + bx, where a = 4 and b = -2. This function represents a cubic polynomial with a leading coefficient of 4.

Graph Plotting:

Let's plot the graph of the function y = 4x^3 - 2x to visualize its behavior and shape.

import matplotlib.pyplot as plt
import numpy as np

# Define the function
def f(x):
    return 4*x**3 - 2*x

# Generate x values
x = np.linspace(-2, 2, 400)
# Calculate corresponding y values
y = f(x)

# Plot the graph
plt.figure(figsize=(8, 6))
plt.plot(x, y, label='$y = 4x^3 - 2x$')
plt.title('Graph of y = 4x^3 - 2x')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.axhline(0, color='black',linewidth=0.5)
plt.axvline(0, color='black',linewidth=0.5)
plt.legend()
plt.show()

The graph of the function y = 4x^3 - 2x is a cubic curve with specific behavior and characteristics based on its polynomial nature.

Feel free to ask if you have further questions or need additional assistance!