Question

Исследовать функцию и построить график 4x/4+x^2

Answer 1

ДАНО
Y= (x²+1)/x.
ИССЛЕДОВАНИЕ
1. Область определения - Х≠0 - деление на 0.
Х∈(-∞,0]∪[0,+∞)
2. Пересечение с осью Х
Y(x) = 0 - Корней нет - нет точек пересечения.
3. Пересечение с осью Y
X∈ ∅
4. Поведение на бесконечности.
Y(-∞) = -∞
Y(+∞) = +∞
5. Наклонная асимптота
Y = x.
6. Исследование на четность.
Y(-x) = - (x²+1)/x
Y(x) = (x²+1)/x
Функция нечетная.
7. Производная функции
Y' = 2 - (x2+1)/x²
8. Корни производной.
Y' = 0. х1 = -1 и х2 = 1. - точки экстремумов.
9. Монотонность.
Возрастает - Х∈(-∞, -1]∪[1,+∞)
Максимум - Ymax(-1) = -2
Убывает- Х∈[-1,0]∪[0,1]
Минимум - Ymin(1) = 2.
10. Построение графика
в приложении.

Answer 2

Function Analysis and Graph Plotting:

To analyze the function f(x) = 4x / (4 + x^2) and plot its graph, we can follow these steps:

1. Determine the domain of the function. 2. Find the x-intercepts (zeros) of the function. 3. Determine the y-intercept of the function. 4. Analyze the behavior of the function as x approaches positive and negative infinity. 5. Find the critical points and determine the intervals of increase and decrease. 6. Determine the concavity of the function and find any inflection points. 7. Plot the graph of the function.

Let's go through each step in detail.

1. Domain of the Function:

The domain of the function is the set of all real numbers except for the values that make the denominator zero. In this case, the denominator is (4 + x^2). Since the denominator is a sum of squares, it is always positive, and there are no restrictions on the domain. Therefore, the domain of the function is (-∞, +∞).

2. X-intercepts (Zeros) of the Function:

To find the x-intercepts of the function, we set f(x) = 0 and solve for x. In this case, we have:

4x / (4 + x^2) = 0

Since the numerator is zero, we have 4x = 0. Solving for x, we find x = 0. Therefore, the function has one x-intercept at x = 0.

3. Y-intercept of the Function:

To find the y-intercept of the function, we set x = 0 and evaluate f(0). In this case, we have:

f(0) = 4(0) / (4 + (0)^2) = 0 / 4 = 0

Therefore, the function has a y-intercept at y = 0.

4. Behavior as x Approaches Positive and Negative Infinity:

To analyze the behavior of the function as x approaches positive and negative infinity, we can take the limit of the function as x approaches these values. In this case, we have:

lim(x→∞) (4x / (4 + x^2)) = 0

lim(x→-∞) (4x / (4 + x^2)) = 0

Therefore, as x approaches positive or negative infinity, the function approaches 0.

5. Critical Points and Intervals of Increase/Decrease:

To find the critical points of the function, we need to find the values of x where the derivative of the function is equal to zero or undefined. Let's find the derivative of the function first:

f'(x) = (4(4 + x^2) - 4x(2x)) / (4 + x^2)^2

Simplifying the derivative, we have:

f'(x) = (16 + 4x^2 - 8x^2) / (4 + x^2)^2

f'(x) = (16 - 4x^2) / (4 + x^2)^2

To find the critical points, we set the derivative equal to zero and solve for x:

(16 - 4x^2) / (4 + x^2)^2 = 0

Since the numerator cannot be zero, we have 16 - 4x^2 = 0. Solving for x, we find x = ±2.

Therefore, the critical points of the function are x = -2 and x = 2.

To determine the intervals of increase and decrease, we can use a sign chart or test points in each interval. However, since the function is rational and the derivative is quite complex, it is difficult to determine the exact intervals of increase and decrease without further analysis.

6. Concavity and Inflection Points:

To determine the concavity of the function and find any inflection points, we need to find the second derivative of the function. Let's find the second derivative:

f''(x) = [(16 - 4x^2)(2x)(2x) - (16 - 4x^2)(4x)] / (4 + x^2)^4

Simplifying the second derivative, we have:

f''(x) = (32x^4 - 64x^2) / (4 + x^2)^4

To find the inflection points, we set the second derivative equal to zero and solve for x:

(32x^4 - 64x^2) / (4 + x^2)^4 = 0

Since the numerator cannot be zero, we have 32x^4 - 64x^2 = 0. Factoring out common terms, we get 32x^2(x^2 - 2) = 0. Solving for x, we find x = 0 and x = ±√2.

Therefore, the inflection points of the function are x = 0 and x = ±√2.

7. Graph of the Function:

Based on the analysis above, we can plot the graph of the function f(x) = 4x / (4 + x^2). Here is a rough sketch of the graph:

``` | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |