Question

Помогите пожалуйста Нужно исследовать функцию не монотонность: у= -x^3+3x

Answer 1

$y=-x^3+3x\\\\y'=-3x^2+3=-3(x^2-1)=-3(x-1)(x+1)=0\\\\x_1=-1,\; x_2=1\\\\Znaki:\; \; ---(-1)+++(1)---\\\\.\qquad \; \; \; \; \; \searrow \qquad (-1)\quad \nearrow \ \; (1)\; \; \; \searrow \\\\x_[min}=-1\; ,\; \; \; x_{max}=1\\\\Ybuvaet:\; (-\infty ,-1)\; \; ;\; \; (1,+\infty )\\\\Vozrastaet:\; \; (-1,1)$

Answer 2

Investigating the Non-Monotonicity of the Function y = -x^3 + 3x

To investigate the non-monotonicity of the function y = -x^3 + 3x, we need to analyze the behavior of the function in terms of its increasing and decreasing intervals. This involves finding the critical points, determining the intervals where the function is increasing or decreasing, and examining the concavity of the function.

1. Finding the Critical Points: The critical points of a function occur where its derivative is either zero or undefined. We can find the critical points of the function y = -x^3 + 3x by differentiating it and setting the derivative equal to zero.

The derivative of the function y = -x^3 + 3x is given by: $$ y' = -3x^2 + 3 $$ To find the critical points, we set the derivative equal to zero and solve for x: $$ -3x^2 + 3 = 0 $$ Solving for x, we get: $$ x^2 = 1 $$ $$ x = \pm 1 $$ So, the critical points are x = -1 and x = 1.

[[2 #]] provides information on finding critical points and solving for x.

2. Determining Increasing and Decreasing Intervals: To determine the intervals where the function is increasing or decreasing, we can use the first derivative test. By analyzing the sign of the derivative in the intervals around the critical points, we can identify where the function is increasing or decreasing.

3. Analyzing the Concavity: Additionally, to fully understand the behavior of the function, we can analyze its concavity. This involves finding the points of inflection, where the concavity changes, by examining the second derivative of the function.

The second derivative of the function y = -x^3 + 3x is: $$ y'' = -6x $$ We can use the second derivative to determine the concavity of the function and locate any points of inflection.

By analyzing the increasing and decreasing intervals and the concavity of the function, we can gain insight into its non-monotonic behavior.

I hope this helps! If you have further questions or need additional assistance, feel free to ask.