Question

Исследовать функцию и построить графикy=4x/(4+x^2) 1)найти область определения функции2)Исследовать функцию на непрерывность,четность периодичность3).исследовать функцию на четность, нечетность4)Исследовать функцию на монотонность и экстремум5)Найти интервалы выпуклости и вогнутости ,точки перегиба 6)найти асимптоты графика функции7)точки пересечения графика с осями координат8)построить график

Answer 1

А в задании нет ошибки, случайно?

иначе если писать ограничения на знаменатель, то выходит что х^2должен быть не равен -4, а такого быть не может.

Или я туплю)

Answer 2

Function Analysis: y = (4x)/(4 + x^2) + 1

To analyze the function y = (4x)/(4 + x^2) + 1, we will explore several aspects, including the domain, continuity, symmetry, monotonicity, extrema, concavity, inflection points, asymptotes, and intersections with the coordinate axes.

1. Domain: The domain of a function is the set of all possible input values (x) for which the function is defined. In this case, the function y = (4x)/(4 + x^2) + 1 is defined for all real numbers except when the denominator (4 + x^2) equals zero. Solving 4 + x^2 = 0, we find that x^2 = -4, which has no real solutions. Therefore, the domain of the function is all real numbers.

2. Continuity: A function is continuous if it has no breaks, jumps, or holes in its graph. The function y = (4x)/(4 + x^2) + 1 is continuous for all real numbers since it is a rational function and the denominator is never zero.

3. Symmetry: To determine the symmetry of the function, we need to check if it is even (symmetric with respect to the y-axis) or odd (symmetric with respect to the origin). For the function y = (4x)/(4 + x^2) + 1, we can substitute -x for x and simplify the expression. If the resulting expression is equal to the original function, then the function is even. If it is equal to the negation of the original function, then the function is odd.

Substituting -x for x, we get y = (4(-x))/(4 + (-x)^2) + 1 = (-4x)/(4 + x^2) + 1. Since this is equal to the original function, y = (4x)/(4 + x^2) + 1, the function is even.

4. Monotonicity and Extrema: To analyze the monotonicity (increasing or decreasing behavior) and extrema (maximum or minimum points) of the function, we need to find its derivative and determine where it is positive or negative.

Taking the derivative of y = (4x)/(4 + x^2) + 1 with respect to x, we get:

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

To find the critical points, we set y' = 0 and solve for x:

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

Simplifying, we have 16 - 4x^2 = 0, which gives x^2 = 4. Solving for x, we find x = -2 and x = 2.

To determine the monotonicity, we can choose test points in the intervals (-∞, -2), (-2, 2), and (2, ∞) and evaluate the sign of y' at those points. Alternatively, we can create a sign chart for y'.

Using the sign chart method, we have:

| Interval | (16 - 4x^2)/(4 + x^2)^2 | |----------------|------------------------| | (-∞, -2) | Positive | | (-2, 2) | Negative | | (2, ∞) | Positive |

From the sign chart, we can conclude that the function is increasing on the interval (-∞, -2) and decreasing on the interval (-2, 2) and (2, ∞).

To find the extrema, we evaluate the function at the critical points and the endpoints of the domain:

- When x = -2, y = (4(-2))/(4 + (-2)^2) + 1 = -8/8 + 1 = -1 + 1 = 0. - When x = 2, y = (4(2))/(4 + 2^2) + 1 = 8/8 + 1 = 1 + 1 = 2.

Therefore, the function has a minimum at (-2, 0) and a maximum at (2, 2).

5. Concavity, Inflection Points, and Intervals of Convexity/Concavity: To analyze the concavity (upward or downward curvature), inflection points, and intervals of convexity/concavity of the function, we need to find its second derivative and determine where it is positive or negative.

Taking the second derivative of y = (4x)/(4 + x^2) + 1 with respect to x, we get:

y'' = [(16 - 4x^2)(2(4 + x^2)) - (16 + 4x^2)(2x(2x))]/(4 + x^2)^4 = [(32 + 8x^2 - 8x^2 - 2x^4) - (32 - 8x^2 + 8x^2 + 2x^4)]/(4 + x^2)^4 = -4x^4/(4 + x^2)^4.

To find the inflection points, we set y'' = 0 and solve for x:

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

Since the numerator is always zero, there are no inflection points.

To determine the intervals of convexity/concavity, we can choose test points in the intervals (-∞, 0), (0, ∞), and create a sign chart for y''.

Using the sign chart method, we have:

| Interval | -4x^4/(4 + x^2)^4 | |----------------|------------------| | (-∞, 0) | Negative | | (0, ∞) | Negative |

From the sign chart, we can conclude that the function is concave down on the entire domain.

6. Asymptotes: To find the asymptotes of the function, we need to analyze its behavior as x approaches positive or negative infinity.

As x approaches positive or negative infinity, the term (4x)/(4 + x^2) approaches zero, and the function approaches y = 1.

Therefore, the function has a horizontal asymptote at y = 1.

7. Intersections with Coordinate Axes: To find the points of intersection with the coordinate axes, we set y = 0 and solve for x:

0 = (4x)/(4 + x^2) + 1.

Simplifying, we have 4x = -4 - x^2, which can be rearranged to x^2 + 4x + 4 = 0. Factoring, we get (x + 2)^2 = 0, which gives x = -2.

Therefore, the function intersects the x-axis at (-2, 0).

To find the y-intercept, we substitute x = 0 into the function:

y = (4(0))/(4 + 0^2) + 1 = 0/4 + 1 = 1.

Therefore, the function intersects the y-axis at (0, 1).

8. Graph: Here is the graph of the function y = (4x)/(4 + x^2) + 1:


Please note that the graph is a rough representation and may not be to scale.

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