Question

Полное исследование функции и ее график y=x^2/(x^2+5)

Answer 1

1) Область определения функции. Точки разрыва функции.
2) Четность или нечетность функции.

y(-x) = y(x), четная функция
3) Периодичность функции.
4) Точки пересечения кривой с осями координат.
Пересечение с осью 0Y
x=0, y=0
Пересечение с осью 0X
y=0

x = 0
5) Исследование на экстремум.
y = x^2/(x^2+5)
Поскольку f(-x)=f(x), то функция является четной.
1. Находим интервалы возрастания и убывания. Первая производная.

или

Находим нули функции. Для этого приравниваем производную к нулю
10x = 0
Откуда:
x1 = 0
(-∞ ;0)(0; +∞)f'(x) < 0f'(x) > 0функция убываетфункция возрастает
В окрестности точки x = 0 производная функции меняет знак с (-) на (+). Следовательно, точка x = 0 - точка минимума.
2. Найдем интервалы выпуклости и вогнутости функции. Вторая производная.

или

Находим корни уравнения. Для этого полученную функцию приравняем к нулю.

Откуда точки перегиба:
x1 = -1.291

f''(x) < 0f''(x) > 0f''(x) < 0функция выпуклафункция вогнутафункция выпукла
6) Асимптоты кривой.

Уравнения наклонных асимптот обычно ищут в виде y = kx + b. По определению асимптоты:

Находим коэффициент k:


Находим коэффициент b:


Получаем уравнение горизонтальной асимптоты:
y = 1

Answer 2

Function and Graph Analysis: y = x^2/(x^2 + 5)

The given function is y = x^2/(x^2 + 5). Let's analyze its properties and graph.

To understand the function and its graph, we can start by examining the numerator and denominator separately.

The numerator is x^2, which is a quadratic function. The graph of y = x^2 is a parabola that opens upward, with the vertex at the origin (0, 0). As x increases or decreases, the value of y increases, indicating that the function is always positive.

The denominator is x^2 + 5, which is also a quadratic function. The graph of y = x^2 + 5 is a parabola that opens upward, with the vertex at (0, 5). As x increases or decreases, the value of y increases, indicating that the function is always positive.

Now, let's consider the entire function y = x^2/(x^2 + 5). Since both the numerator and denominator are always positive, the function is always defined for all real values of x except when the denominator is equal to zero. However, in this case, the denominator x^2 + 5 is never equal to zero because the sum of a positive number (x^2) and another positive number (5) is always positive.

Therefore, the function y = x^2/(x^2 + 5) is defined for all real values of x.

To analyze the behavior of the function as x approaches positive or negative infinity, we can consider the limit of the function as x approaches infinity and negative infinity.

As x approaches positive infinity, both the numerator and denominator of the function approach infinity. Therefore, the function approaches 1, as the ratio of two large positive numbers tends to 1.

As x approaches negative infinity, both the numerator and denominator of the function approach infinity. Therefore, the function also approaches 1, as the ratio of two large positive numbers tends to 1.

Based on the analysis above, we can conclude that the function y = x^2/(x^2 + 5) is always positive and defined for all real values of x. It approaches 1 as x approaches positive or negative infinity.

Now, let's plot the graph of the function to visualize its shape.

Graph of y = x^2/(x^2 + 5)

To plot the graph of the function, we can use a graphing tool or software. Here is the graph of the function:


The graph of the function is a curve that approaches 1 as x approaches positive or negative infinity. It is always positive and defined for all real values of x.

I hope this analysis and graph help you understand the function y = x^2/(x^2 + 5). Let me know if you have any further questions!