Question

Помогите пожалуйста решить пример с пояснением. Найти промежуток выпуклости и вогнутости: y=ln(2+1\x);

Answer 1

Ответ:

y=ln(2+1\x

для начала мы раскрываем скобки,

y=ln • 2+1\x

теперь складываем вместе буквенную сторону

y:ln= 2+1\x

по сути 1\х это просто х так что делаем так:

y:ln =2+x

y:ln= 2x

Answer 2

Problem:

You are given the equation y = ln(2 + 1/x) and you need to find the intervals of convexity and concavity.

Solution:

To find the intervals of convexity and concavity, we need to determine the second derivative of the function and analyze its sign.

Let's start by finding the first derivative of the function:

y' = (1 / (2 + 1/x)) * (-1/x^2)

Now, let's find the second derivative by differentiating the first derivative:

y'' = (-1/x^2) * (-1/x^2) * (2 + 1/x) + (1 / (2 + 1/x)) * (2/x^3)

Simplifying the expression, we get:

y'' = (1 / x^4) * (2 + 1/x) + (2 / (x^3 * (2 + 1/x)))

To analyze the sign of the second derivative, we need to find the critical points where the second derivative is equal to zero or undefined.

Setting the second derivative equal to zero, we have:

(1 / x^4) * (2 + 1/x) + (2 / (x^3 * (2 + 1/x))) = 0

Multiplying through by x^4 * (2 + 1/x), we get:

2 + 1/x + 2 = 0

Simplifying the equation, we have:

3 + 1/x = 0

Subtracting 3 from both sides, we get:

1/x = -3

Taking the reciprocal of both sides, we have:

x = -1/3

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

1. For x < -1/3: - In this interval, both x and (2 + 1/x) are negative. - Therefore, the second derivative is positive. - This means the function is concave up in this interval.

2. For -1/3 < x < 0: - In this interval, x is negative and (2 + 1/x) is positive. - Therefore, the second derivative is negative. - This means the function is concave down in this interval.

3. For x > 0: - In this interval, both x and (2 + 1/x) are positive. - Therefore, the second derivative is positive. - This means the function is concave up in this interval.

Conclusion:

The function y = ln(2 + 1/x) is concave up for x < -1/3 and x > 0, and concave down for -1/3 < x < 0.