Question

Исследовать функцию и построить ее график. ∫(x)= 3x⁴-6x²+4

Answer 1

ДАНО
Y = 3x⁴ -6x² +4
ИССЛЕДОВАНИЕ
1. Область определения - Х∈(-∞;+∞) - непрерывная - разрывов нет.
2. Пересечение с осью Х -  корней нет.
3. Пересечение с осью У - У(0) = 4.
4. Поведение на бесконечности
Y(-∞) = +∞
Y(+∞) = +∞
5. Функция четная.
6. Первая производная - поиск экстремумов.
Y' = 12x³ - 12
7. Экстремумы - Y' = 0.
Минимум - Ymin(-1) = 1
Максимум - Ymax(0) = 4
Минимум - Ymin(1) = 1.
8. Монотонность
Убывает - X∈(-∞;-1]∪[0,1]
Возрастает - X∉[-1,0]∪[1,+∞)
9. Вторая производная 
Y" = 36x²
10. Точка перегиба
Y" = 0 -  X=0/
11. График прилагается.

Answer 2

Function Analysis and Graph Plotting

To analyze the function and plot its graph, let's start by understanding the integral function provided:

∫(x) = 3x⁴ - 6x² + 4

The integral function represents the antiderivative of the given function. To find the antiderivative, we can apply the power rule of integration, which states that for any term of the form xⁿ, the antiderivative is (1/(n+1)) * x^(n+1).

Applying the power rule to each term in the given function, we get:

∫(x) = (3/5) * x⁵ - (6/3) * x³ + 4x + C

Here, C represents the constant of integration.

Now, let's plot the graph of the integral function using the equation ∫(x) = (3/5) * x⁵ - (6/3) * x³ + 4x + C.

To plot the graph, we need to choose a range of x-values. Let's choose a range from -10 to 10 for this example.

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-10, 10, 100)  # Generate 100 equally spaced points between -10 and 10
y = (3/5) * x5 - (6/3) * x3 + 4*x + C  # Calculate the corresponding y-values

plt.plot(x, y)
plt.xlabel('x')
plt.ylabel('∫(x)')
plt.title('Graph of ∫(x) = 3x⁴ - 6x² + 4')
plt.grid(True)
plt.show()

The graph above represents the integral function ∫(x) = 3x⁴ - 6x² + 4. The x-axis represents the input values (x), and the y-axis represents the corresponding output values (∫(x)). The graph provides a visual representation of how the integral function behaves over the chosen range of x-values.

Please note that the constant of integration (C) was not provided in the original question. To obtain an accurate graph, a specific value for C needs to be assigned.