Question

Исследовать функцию! х3 - 3х

Answer 1

1. Область определения функции: множество всех действительных чисел

2. f(-x) = - x³ + 3x = - (x³ - 3x) = - f(x) - нечетная функция

3. Непериодическая.

4. Точки пересечения с осью Ох и Оу.

f(x) = 0;  x³ - 3x = 0;   x(x²-3)=0

(0;0), (-√3;0), (√3;0) - точки пересечения с осью Ох

x=0; f(x)=0;  (0;0) - с осью Оу.

5. Монотонность функции

f'(x) = (x³ - 3x)' = 3x² - 3 = 0

x² - 1 = 0

x = ±1

___+___(-1)___-__(1)__+___
Функция возрастает на промежутке x ∈ (-∞;-1) и x ∈ (1;+∞), а убывает - x ∈ (-1;1). x=-1 - локальный максимум. х=1 - локальный минимум.

7. Точки перегиба
f''(x) = 6x = 0  откуда х=0

Горизонтальных, наклонных и вертикальных асимптот нет.

Answer 2

Function Analysis: f(x) = x^3 - 3x

To analyze the function f(x) = x^3 - 3x, we can examine its properties such as the domain, range, symmetry, intercepts, and behavior.

Domain: The domain of a function represents the set of all possible input values (x-values) for which the function is defined. In this case, since the function f(x) = x^3 - 3x is a polynomial function, it is defined for all real numbers. Therefore, the domain of f(x) is (-∞, ∞).

Range: The range of a function represents the set of all possible output values (y-values) that the function can take. To determine the range of f(x) = x^3 - 3x, we can analyze its behavior as x approaches positive and negative infinity. As x approaches positive infinity, the function also approaches positive infinity. As x approaches negative infinity, the function approaches negative infinity. Therefore, the range of f(x) is (-∞, ∞).

Symmetry: To determine if the function f(x) = x^3 - 3x exhibits any symmetry, we can check if it is an even function (symmetric about the y-axis) or an odd function (symmetric about the origin). For this function, we can observe that f(-x) = (-x)^3 - 3(-x) = -x^3 + 3x = -(x^3 - 3x) = -f(x). Since f(-x) = -f(x), the function is an odd function and exhibits symmetry about the origin.

Intercepts: To find the x-intercepts of the function f(x) = x^3 - 3x, we set f(x) equal to zero and solve for x: x^3 - 3x = 0 x(x^2 - 3) = 0 x = 0 or x = ±√3

Therefore, the x-intercepts of the function are x = 0, x = √3, and x = -√3.

To find the y-intercept, we substitute x = 0 into the function: f(0) = (0)^3 - 3(0) = 0

Therefore, the y-intercept of the function is y = 0.

Behavior: To analyze the behavior of the function f(x) = x^3 - 3x, we can examine its end behavior as x approaches positive and negative infinity. As x approaches positive infinity, the function increases without bound. As x approaches negative infinity, the function decreases without bound. This behavior indicates that the function has no global maximum or minimum.

In summary: - Domain: (-∞, ∞) - Range: (-∞, ∞) - Symmetry: Odd function, symmetric about the origin - X-intercepts: x = 0, x = √3, x = -√3 - Y-intercept: y = 0 - Behavior: Increases without bound as x approaches positive infinity, decreases without bound as x approaches negative infinity.

I hope this analysis helps! Let me know if you have any further questions.