Question

Исследуйте функцию и постройте ее график f(x)=1/4x^4+1

Answer 1

Функция:
симметричная
возрастает от минус бескон. до 0 
убывает от 0 до плюс бесконеч.
максим значение у=1 при х = 0

Answer 2

Function Analysis and Graph Plotting: f(x) = 1/4x^4 + 1

To analyze the function f(x) = 1/4x^4 + 1 and plot its graph, we can start by examining its properties and behavior.

Domain and Range: The function f(x) = 1/4x^4 + 1 is defined for all real numbers x since there are no restrictions on the domain. Therefore, the domain of the function is the set of all real numbers (R). The range of the function depends on the behavior of the function as x approaches positive or negative infinity.

Symmetry: To determine if the function has any symmetry, we can check if f(x) is an even or odd function. An even function satisfies the condition f(-x) = f(x), while an odd function satisfies the condition f(-x) = -f(x).

According to the search result snippet from Source the function f(x) = 1/4x^4 + 1 is not explicitly mentioned. Therefore, we cannot determine its symmetry based on the provided information.

Critical Points and Extrema: To find the critical points and extrema of the function, we need to find the values of x where the derivative of f(x) is equal to zero or undefined. The critical points can be local minima, local maxima, or points of inflection.

The derivative of f(x) = 1/4x^4 + 1 can be found by applying the power rule. According to the search result snippet from Source the derivative of x^4 is 4x^3. Therefore, the derivative of f(x) = 1/4x^4 + 1 is f'(x) = x^3.

To find the critical points, we set f'(x) = 0 and solve for x: x^3 = 0 x = 0

The critical point of the function is x = 0.

Behavior at Infinity: To determine the behavior of the function as x approaches positive or negative infinity, we can examine the leading term of the function. The leading term is the term with the highest power of x.

The leading term of f(x) = 1/4x^4 + 1 is 1/4x^4. As x approaches positive or negative infinity, the leading term dominates the function, and the function approaches positive infinity.

Graph Plotting: Based on the information gathered, we can plot the graph of the function f(x) = 1/4x^4 + 1. Since the function is a polynomial, it is continuous and smooth.

Here is the graph of the function f(x) = 1/4x^4 + 1:


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.