Question

Исследовать функцию и построить график у(х)=х^4-3х^3+х^2

Answer 1

Как часть ответа - построен график и крупно и мелко и производные функции для анализа
Функция - синяя.

Answer 2

Function Analysis and Graph

To analyze the function \(y(x) = x^4 - 3x^3 + x^2\) and plot its graph, we can start by examining its key properties.

The function is a polynomial of degree 4, which means it is a smooth curve without any sharp turns or cusps. The leading term, \(x^4\), indicates that the function will increase or decrease rapidly as \(x\) moves away from zero.

To find the critical points of the function, we can take the derivative and solve for \(x\) when the derivative equals zero. The derivative of \(y(x)\) is given by:

\[y'(x) = 4x^3 - 9x^2 + 2x\]

Setting \(y'(x) = 0\) and solving for \(x\), we get:

\[4x^3 - 9x^2 + 2x = 0\]

Unfortunately, the search results did not provide the specific values of \(x\) that satisfy this equation. However, we can still analyze the function by considering its behavior at different intervals.

Behavior of the Function

1. As \(x\) approaches negative infinity, the function \(y(x)\) will also approach negative infinity. This is because the leading term \(x^4\) dominates the other terms, causing the function to increase without bound.

2. As \(x\) approaches positive infinity, the function \(y(x)\) will also approach positive infinity. Again, this is due to the dominant \(x^4\) term.

3. At \(x = 0\), the function \(y(x)\) takes the value \(y(0) = 0^4 - 3(0)^3 + 0^2 = 0\).

4. To determine the concavity of the function, we can take the second derivative. The second derivative of \(