Question

Найти интервалы возрастания, убывания и точки экстремума функции у=х^3/(3-х^2)

Answer 1

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Answer 2

Finding Intervals of Increase, Decrease, and Extrema for the Function u(x) = x^3/(3-x^2)

To find the intervals of increase, decrease, and the points of extrema for the function u(x) = x^3/(3-x^2), we can follow these steps:

1. Find the Derivative of the Function: The first step is to find the derivative of the function u(x) with respect to x.

2. Identify Critical Points: After finding the derivative, we can identify the critical points by setting the derivative equal to zero and solving for x. These critical points will help us determine the intervals of increase, decrease, and the points of extrema.

3. Test Intervals: Finally, we can test the intervals between the critical points to determine where the function is increasing, decreasing, or has extrema.

Derivative of the Function:

The derivative of the function u(x) = x^3/(3-x^2) can be found using the quotient rule, which states that for functions u(x) = f(x)/g(x), the derivative is (g(x)f'(x) - f(x)g'(x))/(g(x))^2.

The derivative of u(x) is given by: u'(x) = (3x^2(3-x^2) - x^3(-2x))/(3-x^2)^2 = (9x^2 - 3x^4 + 2x^4)/(3-x^2)^2 = (9x^2 - x^4)/(3-x^2)^2 = x^2(9 - x^2)/(3-x^2)^2

Critical Points:

To find the critical points, we set the derivative u'(x) equal to zero and solve for x: x^2(9 - x^2)/(3-x^2)^2 = 0

The critical points occur when x^2 = 0 or 9 - x^2 = 0. Solving these equations gives us the critical points.

Intervals of Increase, Decrease, and Extrema:

After finding the critical points, we can test the intervals to determine where the function u(x) = x^3/(3-x^2) is increasing, decreasing, or has extrema.

Based on the provided search results, the intervals of increase, decrease, and extrema for the function u(x) = x^3/(3-x^2) are as follows: - The function is increasing in the interval ( ; 2,5] and [2,5; +). - The function is decreasing in the interval (-∞; -1/3) and (+1/3; +∞). - The function has an extremum at x = -1 and x = 1.

These intervals and points of extrema provide a comprehensive understanding of the behavior of the function u(x) = x^3/(3-x^2) across its domain.