Question

Исследовать функцию и построить её график : y=3x-x^3

Answer 1

Исследуем с помощью производной.
$Y^{I}=3-3 x^{2}$
3-3x^2=0
-3x^2=-3
x^2=1
x1=1
x2=-1
Выясним знаки производной.
от -бесконечности до -1<0, от -1 до 1 >0, от 1 до +бесконечности <0
При х=-1 min функции, при х=1 max функции
у(-1)=-2
у(1)=2
(-1;-2)
(1;20
Точка пересечения с осью ОУ=0
Точка пересечения с осью ОХ   х1=0, х2=√3,  х3=-√3
Отметь все точки и плавно соедини.

Answer 2

Function Analysis and Graph Plotting: y = 3x - x^3

To analyze the function y = 3x - x^3 and plot its graph, let's break down the process step by step.

Step 1: Determine the Function Type The given function, y = 3x - x^3, is a polynomial function of degree 3. It is a cubic function.

Step 2: Find the y-Intercept To find the y-intercept, we set x = 0 and solve for y: y = 3(0) - (0)^3 y = 0 Therefore, the y-intercept is (0, 0).

Step 3: Find the x-Intercepts To find the x-intercepts, we set y = 0 and solve for x: 0 = 3x - x^3 x(3 - x^2) = 0 x = 0 or x = ±√3 Therefore, the x-intercepts are (0, 0), (√3, 0), and (-√3, 0).

Step 4: Determine the Symmetry To determine the symmetry of the function, we check if it is an even function (symmetric about the y-axis) or an odd function (symmetric about the origin). For the given function, y = 3x - x^3, it is neither even nor odd. Therefore, it does not exhibit any symmetry.

Step 5: Find the Vertical Asymptotes To find the vertical asymptotes, we set the denominator of the function equal to zero and solve for x. However, since the given function does not have a denominator, there are no vertical asymptotes.

Step 6: Find the Horizontal Asymptotes To find the horizontal asymptotes, we examine the behavior of the function as x approaches positive or negative infinity. For a cubic function, there are no horizontal asymptotes.

Step 7: Determine the Increasing and Decreasing Intervals To determine the increasing and decreasing intervals, we find the critical points of the function. The critical points occur where the derivative of the function is equal to zero or undefined.

The derivative of y = 3x - x^3 is given by: y' = 3 - 3x^2

Setting y' = 0, we have: 3 - 3x^2 = 0 3x^2 = 3 x^2 = 1 x = ±1

Therefore, the critical points are x = -1 and x = 1.

We can now determine the intervals of increase and decrease: - For x < -1, y' > 0, so the function is increasing. - For -1 < x < 1, y' < 0, so the function is decreasing. - For x > 1, y' > 0, so the function is increasing.

Step 8: Plot the Graph Now that we have gathered all the necessary information, we can plot the graph of the function y = 3x - x^3.

Here is the graph of the function:


Please note that the graph is a visual representation and may not be perfectly accurate. It provides a general understanding of the shape and behavior of the function.

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