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f(x)=x(x+3)(x+1)(x-4) has zeros at
x=-3,x=-1,x=0, and
x=4.
What is the sign of
f on the interval
-1 < x < 4 ?
Choose 1 answer:
(A)
f is always positive on the interval.
(B)
f is always negative on the interval.
(C)
f is sometimes positive and sometimes negative on the interval.

$f(x)=x(x+3)(x+1)(x-4)$ has zeros at $x=-3, x=-1, x=0$ , and $x=4$ . $\newline$ What is the sign of $f$ on the interval $-1<x<4$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $f$ is always positive on the interval. $\newline$ (B) $f$ is always negative on the interval. $\newline$ (C) $f$ is sometimes positive and sometimes negative on the interval.

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Full solution

Q.

f(x)=x(x+3)(x+1)(x-4)

has zeros at

x=-3, x=-1, x=0

, and

x=4

.

\newline

What is the sign of

f

on the interval

-1<x<4

?

\newline

Choose

1

answer:

\newline

(A)

f

is always positive on the interval.

\newline

(B)

f

is always negative on the interval.

\newline

(C)

f

is sometimes positive and sometimes negative on the interval.

Identify Function Zeros: Since $f(x)$ has zeros at $x=-3$ , $x=-1$ , $x=0$ , and $x=4$ , we know that the function changes sign at each of these points.
Analyze Sign Changes: Between $x=-1$ and $x=0$ , $f(x)$ will have a consistent sign because there are no zeros in that interval.
Positive Region Near $-1$ : $f(x)$ is positive just to the right of $x=-1$ because if we plug in a value slightly greater than $-1$ , all factors of $f(x)$ are positive except $(x+1)$ , which is slightly positive.
Sign Change at $x=0$ : At $x=0$ , $f(x)$ changes sign because $x=0$ is a zero of the function.
Negative Region Near $0$ : Just to the right of $x=0$ , $f(x)$ is negative because if we plug in a value slightly greater than $0$ , the factors $(x+3)$ and $(x+1)$ are positive, but $x$ and $(x-4)$ are slightly negative.
Sign Change at $x=4$ : $f(x)$ remains negative until we reach the next zero at $x=4$ , where it will change sign again.
Interval $-1 < x < 4$ : Therefore, on the interval $-1 < x < 4$ , $f(x)$ is sometimes positive and sometimes negative.

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