Question

Срочно помогите найти вогнутость и выпуклость функции и точки экстремумов (x^2+6*x-4)/(x+3)

Answer 1

1) Область определения  - непрерывность
х+3 ≠ 0 или х≠ -3
2)  Монотонность
Возрастает - х∈(-∞-3]∪[-3;+∞)
График прилагается.

Answer 2

Finding the Concavity and Convexity of a Function

To find the concavity and convexity of a function, as well as the points of extrema, we need to analyze the second derivative of the function. Let's consider the function f(x) = (x^2 + 6x - 4)/(x + 3).

To determine the concavity and convexity of the function, we need to find the second derivative of f(x) and analyze its sign. If the second derivative is positive, the function is concave up (convex) in that interval. If the second derivative is negative, the function is concave down (concave) in that interval.

Let's calculate the second derivative of f(x) and analyze its sign.

Calculating the Second Derivative

To find the second derivative of f(x), we need to differentiate f(x) twice. Let's start by finding the first derivative of f(x):

f'(x) = (2x(x + 3) - (x^2 + 6x - 4))/(x + 3)^2

Simplifying the expression, we get:

f'(x) = (2x^2 + 6x - x^2 - 6x + 4)/(x + 3)^2

f'(x) = (x^2 + 4)/(x + 3)^2

Now, let's differentiate f'(x) to find the second derivative:

f''(x) = [(2x)(x + 3)^2 - (x^2 + 4)(2(x + 3))]/(x + 3)^4

Simplifying the expression, we get:

f''(x) = (2x(x^2 + 6x + 9) - 2(x^2 + 4)(x + 3))/(x + 3)^4

f''(x) = (2x^3 + 12x^2 + 18x - 2x^3 - 8x^2 - 12x - 6x^2 - 24x - 36)/(x + 3)^4

f''(x) = (-14x^2 - 18x - 36)/(x + 3)^4

Analyzing the Sign of the Second Derivative

To analyze the sign of the second derivative, we need to find the critical points and determine the intervals where the second derivative is positive or negative.

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

-14x^2 - 18x - 36 = 0

Unfortunately, the equation above does not factor easily. We can use the quadratic formula to find the solutions for x:

x = (-b ± √(b^2 - 4ac))/(2a)

For our equation, a = -14, b = -18, and c = -36. Plugging these values into the quadratic formula, we get:

x = (-(-18) ± √((-18)^2 - 4(-14)(-36)))/(2(-14))

Simplifying further, we have:

x = (18 ± √(324 - 2016))/(-28)

x = (18 ± √(-1692))/(-28)

Since the discriminant is negative, there are no real solutions for x. Therefore, there are no critical points.

Now, let's analyze the sign of the second derivative in different intervals:

1. For x < -3: - Plugging in a value less than -3 into the second derivative, we get a negative value. - Therefore, the second derivative is negative in this interval, indicating concave down (concave).

2. For x > -3: - Plugging in a value greater than -3 into the second derivative, we get a negative value. - Therefore, the second derivative is negative in this interval, indicating concave down (concave).

Based on the analysis of the second derivative, we can conclude that the function f(x) = (x^2 + 6x - 4)/(x + 3) is concave down (concave) for all values of x.

Points of Extrema

Since the function is concave down (concave) for all values of x, there are no points of extrema.

In summary, the function f(x) = (x^2 + 6x - 4)/(x + 3) is concave down (concave) for all values of x, and there are no points of extrema.

Please let me know if there's anything else I can help you with!