Question

answer parts a.- e. for the function shown below. f(x)=3x^{2}+x^{3} a. use the leading - coefficient test to determine the graphs end behavior. which statement describes the end behavior of f(x)? a. the graph of f(x) rises left and falls right. b. the graph of f(x) rises left and rises right. c. the graph of f(x) falls left and rises right. d. the graph of f(x) falls left and falls right. b. find the x - intercepts. state whether the graph crosses the x - axis, or touches the x - axis and turns around, at each intercept. what are the x - intercepts? x = (use a comma to separate answers as needed.)

Explanation:

Step1: Identify the leading - term

The function is $f(x)=3x^{2}+x^{3}$. The leading - term is $x^{3}$ (the term with the highest power of $x$), and the leading coefficient is $1$ (the coefficient of $x^{3}$).

Step2: Apply the leading - coefficient test

For a polynomial function $y = a_nx^n+\cdots+a_0$ with leading coefficient $a_n$ and degree $n$. Here $n = 3$ (odd) and $a_n=1>0$. When $n$ is odd and $a_n>0$, as $x\to-\infty$, $y\to-\infty$ (the graph falls to the left), and as $x\to+\infty$, $y\to+\infty$ (the graph rises to the right). So the answer to part a is C.

Step3: Find the x - intercepts

Set $f(x)=0$, so $3x^{2}+x^{3}=x^{2}(3 + x)=0$.

Step4: Solve for x

Using the zero - product property, if $x^{2}(3 + x)=0$, then $x^{2}=0$ or $3 + x=0$. Solving $x^{2}=0$ gives $x = 0$ and solving $3 + x=0$ gives $x=-3$.

Step5: Determine the behavior at the x - intercepts

For $x = 0$, the factor $x$ has an even multiplicity ($2$), so the graph touches the $x$ - axis and turns around at $x = 0$. For $x=-3$, the factor $(x + 3)$ has an odd multiplicity ($1$), so the graph crosses the $x$ - axis at $x=-3$.

Answer:

a. C. The graph of f(x) falls left and rises right.
b. $x=-3,0$