Question

Исследовать функцию f(x)=x^5/5-4x^3/3 и построить ее график

Answer 1

ДАНО

Y=1/5*x⁵ - 4/3*x³

1.Область определения D(x) - Х∈(-∞;+∞) - непрерывная.

Вертикальных асимптот - нет.

2. Пересечение с осью Х. Y= x³*(x²/5 - 4/3). Корни: х₁,₂ = +/- 2/3*√15,  х₃ = 0. 

3. Пересечение с осью У.  У(0) = 0. 

4. Поведение на бесконечности.limY(-∞) = - ∞  limY(+∞) = +∞.

Горизонтальной асимптоты - нет. 

5. Исследование на чётность.Y(-x) ≠ Y(x).

Функция ни чётная ни нечётная. 

6. Производная функции.Y'(x)= x⁴ - 4*х² = х²*(х - 2)*(x+2) = 0   . 

Корни: х₁=0 , х₂ = 2, x₃ = -2. 

7. Локальные экстремумы. 

Максимум Ymax(-2)= 64/15 ≈ 4.3, минимум – Ymin(2)= - 64/15 . 

8. Интервалы монотонности.

Возрастает - Х∈[-2;2] , убывает = Х∈(-∞;-2)∪(2;+∞). 

8. Вторая производная - Y"(x) = 4*x*(x - 2)=0. 

Корни производной - точки перегиба  - x₁= 0, x₂ = √2 ≈ 1.4 x₃ = -√2. 

9. Выпуклая “горка» Х∈(-∞;-√2)∪[0.√2], Вогнутая – «ложка» Х∈(-√2;0])∪[√2;+∞). 

10. Область значений Е(у) У∈(-∞;+∞) 

11. Наклонная асимптота. Уравнение: lim(oo)(k*x+b – f(x).  

k=lim(oo)Y(x)/x = ∞.  Наклонной асимптоты - нет

12. График в приложении.

Answer 2

Function Analysis and Graph Plotting

To analyze the function f(x) = (x^5)/5 - (4x^3)/3 and plot its graph, we can follow these steps:

1. Find the derivative of the function to determine critical points and intervals of increase and decrease. 2. Determine the concavity of the function to identify points of inflection. 3. Find the x-intercepts and y-intercept. 4. Plot the graph using the obtained information.

Let's go through each step in detail.

Step 1: Finding the Derivative

To find the derivative of the function f(x) = (x^5)/5 - (4x^3)/3, we can apply the power rule and constant rule of differentiation. The derivative of each term can be calculated as follows:

- The derivative of (x^5)/5 is (5x^4)/5 = x^4. - The derivative of (4x^3)/3 is (4 * 3x^2)/3 = 4x^2.

Therefore, the derivative of the function is f'(x) = x^4 - 4x^2.

Step 2: Determining Critical Points and Intervals of Increase/Decrease

To find the critical points, we need to solve the equation f'(x) = 0. Let's solve it:

x^4 - 4x^2 = 0

Factoring out x^2, we get:

x^2(x^2 - 4) = 0

This equation has two solutions:

- x^2 = 0 gives us x = 0. - x^2 - 4 = 0 gives us x = 2 and x = -2.

Now, we can determine the intervals of increase and decrease by analyzing the sign of the derivative in different intervals:

- For x < -2, f'(x) < 0, so the function is decreasing. - For -2 < x < 0, f'(x) > 0, so the function is increasing. - For 0 < x < 2, f'(x) < 0, so the function is decreasing. - For x > 2, f'(x) > 0, so the function is increasing.

Step 3: Determining Concavity and Points of Inflection

To determine the concavity of the function, we need to find the second derivative. Let's find it:

Taking the derivative of f'(x) = x^4 - 4x^2, we get:

f''(x) = 4x^3 - 8x

Now, we can find the points of inflection by solving the equation f''(x) = 0:

4x^3 - 8x = 0

Factoring out 4x, we get:

4x(x^2 - 2) = 0

This equation has two solutions:

- 4x = 0 gives us x = 0. - x^2 - 2 = 0 gives us x = √2 and x = -√2.

Now, we can determine the concavity of the function by analyzing the sign of the second derivative in different intervals:

- For x < -√2, f''(x) < 0, so the function is concave down. - For -√2 < x < 0, f''(x) > 0, so the function is concave up. - For 0 < x < √2, f''(x) < 0, so the function is concave down. - For x > √2, f''(x) > 0, so the function is concave up.

Step 4: Finding Intercepts and Plotting the Graph

To find the x-intercepts, we need to solve the equation f(x) = 0:

(x^5)/5 - (4x^3)/3 = 0

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

f(0) = (0^5)/5 - (4 * 0^3)/3 = 0

Now, let's plot the graph of the function using the obtained information:


Please note that the graph is a visual representation and may not be to scale. The x-intercepts, y-intercept, critical points, and points of inflection can be accurately determined from the calculations.

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