Question
Introduce your twice-differentiable function, for example:f(x)=x^3−3x^2+2.
Answers
Answered by
GPT 3.5
I would like to introduce the function f(x) = x^3 - 3x^2 + 2. This is a twice-differentiable function, which means that it can be differentiated twice to obtain the second derivative.
The function is a polynomial function of degree three. It consists of three terms: x^3, -3x^2, and 2. The first term, x^3, represents the cubic behavior of the function. The second term, -3x^2, represents the quadratic nature of the function. And the constant term, 2, simply adds a vertical shift to the graph of the function.
To find the first derivative of f(x), we can differentiate each term separately using the power rule for differentiation. The derivative of x^3 is 3x^2, the derivative of -3x^2 is -6x, and the derivative of the constant term 2 is 0. Therefore, the first derivative of f(x) is f'(x) = 3x^2 - 6x.
To find the second derivative of f(x), we can differentiate the first derivative. Applying the power rule again, the second derivative of f(x) is f''(x) = 6x - 6.
From the second derivative, we can see that the function is a linear function, indicating that the concavity of the graph changes at x = 1. When x < 1, the graph is concave up, and when x > 1, the graph is concave down. This information can help us analyze the behavior and shape of the graph of the function.
Overall, the function f(x) = x^3 - 3x^2 + 2 is a twice-differentiable function that exhibits cubic and quadratic behaviors. Its first derivative is 3x^2 - 6x, and its second derivative is 6x - 6.
The function is a polynomial function of degree three. It consists of three terms: x^3, -3x^2, and 2. The first term, x^3, represents the cubic behavior of the function. The second term, -3x^2, represents the quadratic nature of the function. And the constant term, 2, simply adds a vertical shift to the graph of the function.
To find the first derivative of f(x), we can differentiate each term separately using the power rule for differentiation. The derivative of x^3 is 3x^2, the derivative of -3x^2 is -6x, and the derivative of the constant term 2 is 0. Therefore, the first derivative of f(x) is f'(x) = 3x^2 - 6x.
To find the second derivative of f(x), we can differentiate the first derivative. Applying the power rule again, the second derivative of f(x) is f''(x) = 6x - 6.
From the second derivative, we can see that the function is a linear function, indicating that the concavity of the graph changes at x = 1. When x < 1, the graph is concave up, and when x > 1, the graph is concave down. This information can help us analyze the behavior and shape of the graph of the function.
Overall, the function f(x) = x^3 - 3x^2 + 2 is a twice-differentiable function that exhibits cubic and quadratic behaviors. Its first derivative is 3x^2 - 6x, and its second derivative is 6x - 6.
There are no human answers yet. A form for humans to post answers is coming very soon!