Question

ПОМОГИТЕ ПОЖАЛУЙСТА СРОЧНО НУЖНО ПРОШУ!!! Исследовать функцию и построить график: 1)f(x)=х4-7х2+12 2) f(x)=x4-6x2+8

Answer 1

Ответ:

Объяснение:

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Answer 2

Function Analysis and Graph Plotting

Let's analyze and plot the given functions:

1. Function 1: - The function is: f(x) = x^4 - 7x^2 + 12

2. Function 2: - The function is: f(x) = x^4 - 6x^2 + 8

Analysis of Function 1:

To analyze the function f(x) = x^4 - 7x^2 + 12, we can start by finding its critical points, which are the points where the derivative is zero.

The derivative of the function f(x) with respect to x is given by: f'(x) = 4x^3 - 14x

To find the critical points, we set the derivative equal to zero and solve for x: 4x^3 - 14x = 0 Factor out x: x(4x^2 - 14) = 0 Solving for x, we get x = 0 and x = ±√(14/4) = ±√(7/2)

Now, we can use the second derivative test to determine the nature of these critical points.

The second derivative of f(x) is: f''(x) = 12x^2 - 14

Plugging the critical points into the second derivative, we find: f''(0) = -14 (indicating a local maximum) f''(√(7/2)) = 12(7/2) - 14 = 21 - 14 = 7 (indicating a local minimum) f''(-√(7/2)) = 12(7/2) - 14 = 21 - 14 = 7 (indicating a local minimum)

So, the function has a local maximum at x = 0 and local minima at x = ±√(7/2).

Analysis of Function 2:

For the function f(x) = x^4 - 6x^2 + 8, we can follow a similar process to analyze it.

The derivative of the function f(x) with respect to x is given by: f'(x) = 4x^3 - 12x

Setting the derivative equal to zero and solving for x, we find the critical points at x = 0 and x = ±√(12/4) = ±√3.

Using the second derivative test, we find: f''(0) = 0 (indicating a possible point of inflection) f''(√3) = 12(3) - 12 = 36 - 12 = 24 (indicating a local minimum) f''(-√3) = 12(3) - 12 = 36 - 12 = 24 (indicating a local minimum)

So, the function has local minima at x = ±√3 and a possible point of inflection at x = 0.

Graph Plotting:

Let's plot the graphs of both functions to visualize their behavior.

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-3, 3, 400)
y1 = x4 - 7*x2 + 12
y2 = x4 - 6*x2 + 8

plt.figure(figsize=(10, 6))
plt.plot(x, y1, label='f(x) = x^4 - 7x^2 + 12')
plt.plot(x, y2, label='f(x) = x^4 - 6x^2 + 8')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.title('Graphs of the Given Functions')
plt.legend()
plt.grid(True)
plt.show()

The graph will visually represent the behavior of the functions f(x) = x^4 - 7x^2 + 12 and f(x) = x^4 - 6x^2 + 8, showing their critical points, minima, and maxima.

I hope this analysis and graph plotting help! If you have any further questions or need additional assistance, feel free to ask.