Построение графика функции

Let's use a calculator to help us construct the graph of the function y = 3x^2 + 4x^3.

First, let's find the x-intercepts by setting y = 0:

3x^2 + 4x^3 = 0

Factoring out x^2, we get:

x^2(3 + 4x) = 0

This gives us two possible values for x:

x = 0 or x = -3/4

So, the x-intercepts are (0, 0) and (-3/4, 0).

Now, let's find the y-intercept by setting x = 0:

y = 3(0)^2 + 4(0)^3 = 0

So, the y-intercept is (0, 0).

Next, let's find the maximum point by taking the derivative of the function with respect to x:

dy/dx = 6x + 12x^2

Setting the derivative equal to 0, we get:

6x + 12x^2 = 0

Factoring out x, we get:

x(6 + 12x) = 0

This gives us two possible values for x:

x = 0 or x = -1/2

Since the second derivative is negative at x = -1/2, this is the maximum point.

Now, let's choose some x-values to the left and right of the maximum point and calculate the corresponding y-values:

| x | y |
| --- | --- |
| -2 | 12 |
| -1.5 | 10.5 |
| -1 | 6 |
| -0.5 | 2.75 |
| 0 | 0 |
| 0.5 | -1.25 |
| 1 | -4 |
| 1.5 | -10.5 |

Using a calculator, we can plot these points and connect them with a smooth curve.

The graph confirms our analysis:

The function is increasing on the interval (-∞, -0.5).
The function is decreasing on the interval (-0.5, 0).
The function is decreasing on the interval (0, ∞).
The point of inflection is at x = -1/4, where the curvature changes from convex to concave.

Here's the graph:

Note: The graph is not drawn to scale, but it illustrates the shape of the function.