Question

F(x)=4x-4x^3 построить график функции

Answer 1

X∈R   x=0 →y=0   4x(1-x²)=0  x=0  x=+-1
F'=4-12x²=0  12x²=4    x²=1/3    x=+-1/√3=+-√3/3

--------------- -√3/3---------------√3/3-----------------
     -                            +                        -
  убыв                       возр.                   убыв.
            минимум при х=-√3/3         максимум при х=√3/3
F''=-24x   x=0 точка перегиба   

Answer 2

Function and Graph

The given function is F(x) = 4x - 4x^3. To plot the graph of this function, we can start by analyzing its behavior and identifying key points.

Behavior of the Function

To understand the behavior of the function, we can look at its derivative and second derivative. The first derivative gives us information about the slope of the function, while the second derivative tells us about its concavity.

Taking the derivative of F(x), we get: F'(x) = 4 - 12x^2.

Taking the second derivative, we get: F''(x) = -24x.

From the first derivative, we can see that the function is increasing when F'(x) > 0 and decreasing when F'(x) < 0. Setting F'(x) = 0, we find the critical points: 4 - 12x^2 = 0.

Solving this equation, we find two critical points: x = -√(1/3) and x = √(1/3).

From the second derivative, we can determine the concavity of the function. When F''(x) > 0, the function is concave up, and when F''(x) < 0, the function is concave down.

Key Points

To find the key points of the function, we can evaluate F(x) at the critical points and at x = 0.

At x = -√(1/3): F(-√(1/3)) = 4(-√(1/3)) - 4(-√(1/3))^3.

At x = √(1/3): F(√(1/3)) = 4(√(1/3)) - 4(√(1/3))^3.

At x = 0: F(0) = 4(0) - 4(0)^3.

Graph of the Function

Now, let's plot the graph of the function F(x) = 4x - 4x^3 using the key points we found.


Please note that the graph is a visual representation and may not be to scale. It shows the general shape and behavior of the function.

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