Which Statement Best Describes The Characteristics Of The Graph Of Y = − X 3 + 3 X 2 + 1 Y=-x^3+3x^2+1 Y = − X 3 + 3 X 2 + 1 ?A. Increase: ( − ∞ , 1 ) ∪ ( 2 , ∞ (-\infty, 1) \cup (2, \infty ( − ∞ , 1 ) ∪ ( 2 , ∞ ] And Decrease: ( − 4 , 8 (-4, 8 ( − 4 , 8 ] B. Increase: ( − 1 , 0 (-1, 0 ( − 1 , 0 ] And Decrease: $(-\infty, -1) \cup

Introduction

In mathematics, a cubic function is a polynomial function of degree three, which means the highest power of the variable is three. The graph of a cubic function can have various characteristics, including increasing and decreasing intervals, local maxima and minima, and inflection points. In this article, we will analyze the characteristics of the graph of the cubic function $y=-x^3+3x^2+1$.

The Function and Its Derivative

The given function is $y=-x^3+3x^2+1$. To analyze its characteristics, we need to find its derivative. The derivative of a function is a measure of how the function changes as its input changes. The derivative of the given function is:

$$\frac{dy}{dx}=-3x^2+6x$$

Finding Critical Points

Critical points are the points on the graph where the function changes from increasing to decreasing or vice versa. To find the critical points, we need to find the values of $x$ where the derivative is equal to zero or undefined. Setting the derivative equal to zero, we get:

$$-3x^2+6x=0$$

Solving for $x$, we get:

$$x=0, x=2$$

These are the critical points of the function.

Finding Inflection Points

Inflection points are the points on the graph where the function changes from concave up to concave down or vice versa. To find the inflection points, we need to find the values of $x$ where the second derivative is equal to zero or undefined. The second derivative of the function is:

$$\frac{d^2y}{dx^2}=-6x+6$$

Setting the second derivative equal to zero, we get:

$$-6x+6=0$$

Solving for $x$, we get:

$$x=1$$

This is the inflection point of the function.

Analyzing the Graph

Now that we have found the critical points and inflection point, we can analyze the graph of the function. The graph of the function is a cubic function, which means it can have various characteristics, including increasing and decreasing intervals, local maxima and minima, and inflection points.

Increasing and Decreasing Intervals

To determine the increasing and decreasing intervals of the function, we need to analyze the sign of the derivative. If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing.

For $x<-1$, the derivative is negative, so the function is decreasing.

For $-1<x<0$, the derivative is positive, so the function is increasing.

For $x>0$, the derivative is negative, so the function is decreasing.

For $x>2$, the derivative is positive, so the function is increasing.

Local Maxima and Minima

Local maxima and minima are the points on the graph where the function has a maximum or minimum value. To find the local maxima and minima, we need to analyze the sign of the derivative and the second derivative.

For $x<-1$, the derivative is negative, so the function is decreasing. The second derivative is positive, so the function is concave up.

For $-1<x<0$, the derivative is positive, so the function is increasing. The second derivative is negative, so the function is concave down.

For $x>0$, the derivative is negative, so the function is decreasing. The second derivative is positive, so the function is concave up.

For $x>2$, the derivative is positive, so the function is increasing. The second derivative is negative, so the function is concave down.

Conclusion

In conclusion, the graph of the cubic function $y=-x^3+3x^2+1$ has various characteristics, including increasing and decreasing intervals, local maxima and minima, and inflection points. The function is decreasing for $x<-1$ and $x>0$, increasing for $-1<x<0$ and $x>2$, and concave up for $x<-1$ and $x>0$, and concave down for $-1<x<0$ and $x>2$. The function has a local maximum at $x=0$ and a local minimum at $x=2$.

Answer

Based on the analysis of the graph, the correct answer is:

$$$$

Q: What is a cubic function?

A: A cubic function is a polynomial function of degree three, which means the highest power of the variable is three. The general form of a cubic function is $y=ax^3+bx^2+cx+d$, where $a$, $b$, $c$, and $d$ are constants.

Q: What are the characteristics of a cubic function graph?

A: The graph of a cubic function can have various characteristics, including increasing and decreasing intervals, local maxima and minima, and inflection points. The graph can also have asymptotes, holes, and other features.

Q: How do I find the critical points of a cubic function?

A: To find the critical points of a cubic function, you need to find the values of $x$ where the derivative is equal to zero or undefined. The derivative of a cubic function is a quadratic function, which can be factored or solved using the quadratic formula.

Q: What is an inflection point?

A: An inflection point is a point on the graph where the function changes from concave up to concave down or vice versa. To find the inflection point of a cubic function, you need to find the value of $x$ where the second derivative is equal to zero or undefined.

Q: How do I determine the increasing and decreasing intervals of a cubic function?

A: To determine the increasing and decreasing intervals of a cubic function, you need to analyze the sign of the derivative. If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing.

Q: What is a local maximum or minimum?

A: A local maximum or minimum is a point on the graph where the function has a maximum or minimum value. To find the local maxima and minima of a cubic function, you need to analyze the sign of the derivative and the second derivative.

Q: How do I find the local maxima and minima of a cubic function?

A: To find the local maxima and minima of a cubic function, you need to analyze the sign of the derivative and the second derivative. If the derivative is positive and the second derivative is negative, the function is increasing and concave down, indicating a local maximum. If the derivative is negative and the second derivative is positive, the function is decreasing and concave up, indicating a local minimum.

Q: What is the significance of the inflection point?

A: The inflection point is significant because it indicates a change in the concavity of the function. The inflection point can also be used to determine the local maxima and minima of the function.

Q: How do I graph a cubic function?

A: To graph a cubic function, you need to plot the function on a coordinate plane. You can use a graphing calculator or a computer program to graph the function. You can also use the characteristics of the function, such as the increasing and decreasing intervals, local maxima and minima, and inflection points, to graph the function.

Q: What are some common applications of cubic functions?

A: Cubic functions have many applications in mathematics, science, and engineering. Some common applications include modeling population growth, describing the motion of objects, and analyzing the behavior of complex systems.

Q: How do I solve cubic equations?

A: Cubic equations can be solved using various methods, including factoring, the quadratic formula, and numerical methods. The solution to a cubic equation can be a real number, a complex number, or a combination of both.

Q: What are some common mistakes to avoid when working with cubic functions?

A: Some common mistakes to avoid when working with cubic functions include:

• Not checking the domain of the function
• Not analyzing the sign of the derivative and the second derivative
• Not identifying the inflection point
• Not graphing the function correctly
• Not using the correct methods to solve cubic equations

By understanding the characteristics of a cubic function graph and avoiding common mistakes, you can effectively work with cubic functions and apply them to real-world problems.