Question

question 10 of 10
the function ( f(x) = x^3 - 3x^2 + 2x ) rises as x grows very small.

a. true

b. false

Explanation:

Step1: Analyze the leading term

For a polynomial function \( f(x) = a_nx^n + a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0 \), the end - behavior is determined by the leading term \( a_nx^n \) (the term with the highest power of \( x \)). In the function \( f(x)=x^{3}-3x^{2}+2x \), the leading term is \( x^{3} \) with \( n = 3 \) (odd) and \( a_n=1>0 \).

Step2: Determine the end - behavior for \( x

ightarrow-\infty \)
When \( n \) is odd and \( a_n>0 \), as \( x
ightarrow-\infty \) (x grows very small, i.e., approaches negative infinity), we consider the behavior of \( y = x^{3} \). Let's take values of \( x \) that are very small (negative and with large magnitude), for example, if \( x=-1000 \), then \( x^{3}=(-1000)^{3}=- 10^{9} \), if \( x = - 1001 \), then \( x^{3}=(-1001)^{3}\approx - 10^{9}-3\times10^{6}-3\times10^{3}-1\), which is more negative than \( (-1000)^{3} \). Wait, no, wait: the leading term is \( x^{3} \), and for the function \( y = x^{3} \), when \( x
ightarrow-\infty \), \( y=x^{3}
ightarrow-\infty \), but wait, let's re - examine. Wait, the function \( f(x)=x^{3}-3x^{2}+2x=x(x^{2}-3x + 2)=x(x - 1)(x - 2) \). But for end - behavior, we focus on the leading term. The leading term is \( x^{3} \). The general rule for end - behavior of polynomials:

• If the degree \( n \) is odd:
• If the leading coefficient \( a_n>0 \), as \( x

ightarrow-\infty \), \( f(x)
ightarrow-\infty \) and as \( x
ightarrow+\infty \), \( f(x)
ightarrow+\infty \).

• If the leading coefficient \( a_n<0 \), as \( x

ightarrow-\infty \), \( f(x)
ightarrow+\infty \) and as \( x
ightarrow+\infty \), \( f(x)
ightarrow-\infty \).

In our case, the degree \( n = 3 \) (odd) and leading coefficient \( a_n = 1>0 \). So as \( x
ightarrow-\infty \) (x grows very small), \( f(x)=x^{3}-3x^{2}+2x\approx x^{3}
ightarrow-\infty \), which means the function falls (decreases) as \( x \) grows very small. So the statement "The function \( f(x)=x^{3}-3x^{2}+2x \) rises as \( x \) grows very small" is false.

Answer:

B. False