Question

Иследовать функцию 10 класс. исследовать функцию и построить график -1/3x^3+4x+3

Answer 1

смотри во влажениях. под цифрой 2 еще нужно написать что функция является ни четной ни нечетной

Answer 2

Function Analysis: -1/3x^3 + 4x + 3

To analyze the function f(x) = -1/3x^3 + 4x + 3 and plot its graph, we can examine its key properties such as the domain, range, critical points, and behavior at the extremes. Let's break down each aspect:

Domain and Range

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. The range, on the other hand, represents the set of all possible output values (y-values) that the function can produce.

To determine the domain and range of the given function, we need to consider any restrictions or limitations on the variable x. In this case, there are no apparent restrictions, so the domain is all real numbers. As for the range, since the function is a polynomial, it is also all real numbers.

Critical Points

Critical points are the values of x where the derivative of the function is either zero or undefined. They can help us identify local extrema (maximum or minimum points) and inflection points.

To find the critical points of the function, we need to take the derivative of f(x) with respect to x and solve for x when the derivative equals zero:

f'(x) = -x^2 + 4

Setting f'(x) equal to zero and solving for x:

-x^2 + 4 = 0 x^2 = 4 x = ±2

Therefore, the critical points of the function occur at x = -2 and x = 2.

Behavior at Extremes

To determine the behavior of the function at the extremes, we can evaluate the function at the critical points and at the endpoints of the domain.

Let's calculate the corresponding y-values for the critical points and the endpoints:

f(-2) = -1/3(-2)^3 + 4(-2) + 3 = -8/3 - 8 + 3 = -8/3 - 24/3 + 9/3 = -23/3 f(2) = -1/3(2)^3 + 4(2) + 3 = -8/3 + 8 + 3 = -8/3 + 24/3 + 9/3 = 25/3 f(-∞) = -∞ f(∞) = ∞

Therefore, at x = -2, the function has a local minimum with a y-value of -23/3, and at x = 2, the function has a local maximum with a y-value of 25/3. As x approaches negative or positive infinity, the function approaches negative or positive infinity, respectively.

Graph of the Function

Now, let's plot the graph of the function f(x) = -1/3x^3 + 4x + 3:


Please note that the graph is a visual representation of the function and may not be perfectly accurate due to the limitations of the text-based format.

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