Question

Исследовать функцию у=х+х3-3 и построить ее график

Answer 1

У=4х-3 ...............

Answer 2

Function Analysis and Graph Plotting

To analyze the function f(x) = x^3 - 3x and plot its graph, we can start by examining its key properties.

1. Domain and Range: - The domain of the function is all real numbers since there are no restrictions on the input variable x. - The range of the function is also all real numbers since the function is a polynomial of degree 3, which means it can take any real value.

2. Symmetry: - The function f(x) = x^3 - 3x is an odd function because it satisfies the property f(-x) = -f(x). This means that the graph of the function is symmetric with respect to the origin.

3. Intercepts: - To find the x-intercepts, we set f(x) = 0 and solve for x: ``` x^3 - 3x = 0 x(x^2 - 3) = 0 ``` This equation has three solutions: x = 0 and x = ±√3. Therefore, the function intersects the x-axis at these points. - To find the y-intercept, we substitute x = 0 into the function: ``` f(0) = 0^3 - 3(0) = 0 ``` So, the y-intercept is at the origin (0, 0).

4. Critical Points: - To find the critical points, we take the derivative of the function and set it equal to zero: ``` f'(x) = 3x^2 - 3 = 0 ``` Solving this equation, we find two critical points: x = -1 and x = 1.

5. Behavior at Infinity: - As x approaches positive or negative infinity, the function also approaches positive or negative infinity, respectively. This is because the leading term x^3 dominates the function's behavior.

Now, let's plot the graph of the function f(x) = x^3 - 3x:

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Please note that the graph is a visual representation of the function and can help us understand its behavior and key properties.