Question

F(x)=x^4-4x^3-8x^2+1​

Answer 1

План наших действий:
1) ищем производную
2) приравниваем её к нулю и решаем уравнение (ищем точки экстремума)
3) ставим эти найденные корни на числовой прямой и смотрим знаки производной на каждом промежутке
4) думаем...
5) пишем ответ
Поехали?
f(x) = x^4 - 4x^3 - 8x^2 + 12
1) f'(x) = 4x³ -`12x² -16x
2) 4x³ -`12x² -16x = 0
x(4x² -`12x -16) = 0
x = 0 или 4x² -`12x -16 = 0
х² -3х - 4 = 0
по т. Виета корни 4 и -1
3) -∞ -1 0 4 +∞
- - + + это знаки "х"
+ - - + это знаки 4x² -`12x -16
- + - + это знаки производной
4)Где производная с минусом - там функция "ползёт" вверх(возрастает) где производная с плюсом, там функция "уползает" вниз( убывает)
Если при переходе через точку экстремума производная меняет знак с "-" на "+", то эта точка - точка минимума;
если при переходе через точку экстремума производная меняет знак с "+" на "-", то эта точка - точка максимума;
5) а) f(x) = x^4 - 4x^3 - 8x^2 + 12 возрастает при х∈(-1;0)∪(4;+∞)
f(x) = x^4 - 4x^3 - 8x^2 + 12 убывает при х∈(-∞; -1)∪(0;4)
б) х = -1 это точка минимума; х = 0 - это точка максимума; х = 4 это точка минимума

Answer 2

You've provided a function, F(x), which is a polynomial. To work with this function, we can analyze it and potentially find its roots (the values of x for which F(x) = 0), local extrema, and perform other relevant mathematical operations. Let's get started:

The given function is: F(x) = x^4 - 4x^3 - 8x^2 + 1.

1. To find the roots of the function (the values of x for which F(x) = 0), we can try to factor it or use numerical methods. Factoring the polynomial in this case is not straightforward, so we can resort to numerical methods or graphing. I'll use a numerical method for the sake of illustration.

2. You can use a numerical method like the Newton-Raphson method, a graphing calculator, or software like Python with libraries like NumPy to find the approximate roots of the function. I'll provide a simple Python code snippet to find approximate roots using NumPy:

pythonCopy code
import numpy as np
from scipy.optimize import root_scalar

def f(x):
    return x**4 - 4*x**3 - 8*x**2 + 1

# Use the root_scalar function to find roots
root_result = root_scalar(f, bracket=[-10, 10])
print("Approximate root:", root_result.root)

Keep in mind that this code will find one root within the specified bracket. You may need to change the bracket or run the code multiple times to find all roots.

3. To find local extrema (maximum and minimum points), you can take the derivative of the function F(x) and set it equal to zero to find critical points. Then, you can use the second derivative test to determine whether these critical points are maxima, minima, or saddle points. Here's a general outline of the process:

   a. Find the first derivative of F(x): F'(x) = 4x^3 - 12x^2 - 16x

   b. Find the critical points by setting F'(x) = 0: 4x^3 - 12x^2 - 16x = 0

   c. Solve for x to find the critical points.

   d. Evaluate the second derivative F''(x) = 12x^2 - 24x - 16 at each critical point to classify them as maxima, minima, or saddle points.

The above steps should help you analyze the given function and find its roots and critical points. Depending on the specific values of x, you can then determine the behavior of the function in different regions.