Question

Решите функцию y=1/3x^3-4x

Answer 1

Приравнять функцию к 0 и решаем уравнение

Answer 2

To solve the function y = (1/3)x^3 - 4x, there are a few things we can do. One is to find the derivative of the function, which will give us the slope of the function at any given point. Another is to find the x-intercepts and y-intercepts of the function, which will help us graph it. Let's go through each step:

1. Finding the derivative:

To find the derivative of y = (1/3)x^3 - 4x, we can use the power rule of differentiation. The power rule states that if y = x^n, then the derivative of y with respect to x is dy/dx = nx^(n-1). Applying this rule to our function, we get:

dy/dx = (1/3) * 3x^2 - 4

Simplifying this expression, we get:

dy/dx = x^2 - 4

So the slope of the function at any point is given by the expression x^2 - 4.

2. Finding the x-intercepts:

To find the x-intercepts of the function, we need to set y = 0 and solve for x. So we have:

0 = (1/3)x^3 - 4x

Multiplying both sides by 3, we get:

0 = x^3 - 12x

Factoring out an x, we get:

0 = x(x^2 - 12)

So the x-intercepts are x = 0 and x = +/- sqrt(12) ≈ +/- 3.464.

3. Finding the y-intercept:

To find the y-intercept of the function, we need to set x = 0 and evaluate y. So we have:

y = (1/3) * 0^3 - 4 * 0

So the y-intercept is y = 0.

4. Graphing the function:

Using the information we've gathered so far, we can graph the function as shown below:

yamlCopy code
  |                     
5 |-                    
  |             o       
4 |-                    
  |                      
3 |-             o    
  |                 
2 |-         o           
  |                 
1 |-                      
  |                 
0 |-                      
  |              y = (1/3)x^3 - 4x
-1|-
  |
-2|-
  |
-3|-----------o-----o-
  |     -3.464   0  3.464
  |
  |                      

As we can see from the graph, the function has a local minimum at x = -sqrt(12), a local maximum at x = sqrt(12), and passes through the origin. The slope of the function increases as x moves away from 0, and the function becomes steeper as x gets larger.