Question

Исследовать функцию y=x²+9/x+1 1.Область определения функции 2.Точки пересечения с осями координат 3.Промежутки монотоности и точки экстриума. 4.Асиптоты

Answer 1

Исследование функции Y = X³+6X²+9X.
1. Область определения
Х€(-∞,+∞)
2. Пересечение с осью Х.
Х= 0,  Х = -3.
3. Пересечение с осью У.
У(0) = 0.
4. Поведение на бесконечности.
У(-∞) = -∞
У(+∞) = +∞
5. Исследование на четность.
Y(+x) = x³+6x²+9
Y(-х) = - х³+6х-9
Функция ни четная ни нечетная.
6.  Монотонность.
Производная функции
Y' = 3x²+12x+9
Точки экстремумов
х1 = -3     х2 = -1.
 Ymax(-3) = 0
Ymin(1) = 4.
Возрастает Х€(-∞,-3]∪[-1,+∞)
Убывает X€[-3,-1]
7. Точки перегиба - нули второй производной.
Y" = 6x+12 = 0
Х= -2.
Выпуклая - "горка" - Х€(-∞;-2]
Вогнутая - "ложка" - Х€[-2;+∞)

Answer 2

Function Analysis: y = x² + 9/x + 1 + 1

1. Domain of the Function: - The domain of the function y = x² + 9/x + 1 + 1 is all real numbers except x = -1, as the function is undefined at this point due to division by zero. - The domain can be expressed as: D = (-∞, -1) U (-1, ∞). - The function is defined for all real numbers except -1 .

2. Intercepts with the Coordinate Axes: - x-Intercept: To find the x-intercept, set y = 0 and solve for x. There may be no real solutions for this function. - y-Intercept: To find the y-intercept, set x = 0 and solve for y. The y-intercept is the point (0, 10) as y = 0² + 9/0 + 1 + 1 = 10. - The function intersects the y-axis at the point (0, 10) .

3. Monotonic Intervals and Extrema: - Monotonic Intervals: To determine the intervals of monotonicity, we need to find the critical points of the function. The derivative of the function can help identify these points. - Extrema: Extrema occur at critical points or at the endpoints of the domain. To find the extrema, we need to analyze the critical points and the behavior of the function at the endpoints of the domain.

4. Asymptotes: - Vertical Asymptote: A vertical asymptote occurs at x = -1, as the function becomes undefined at this point due to division by zero. - Horizontal Asymptote: To find the horizontal asymptote, we can analyze the behavior of the function as x approaches positive or negative infinity.

Let's further analyze the monotonic intervals and extrema to determine the function's behavior in different intervals.