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f(x)=(3x-2)(x+4)(9x+5) has zeros at
x=-4,x=-(5)/(9), and
x=(2)/(3).
What is the sign of
f on the interval
-(5)/(9) < x < (2)/(3) ?
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.

$f(x)=(3 x-2)(x+4)(9 x+5)$ has zeros at $x=-4, x=-\frac{5}{9}$ , and $x=\frac{2}{3}$ . $\newline$ What is the sign of $f$ on the interval $-\frac{5}{9}<x<\frac{2}{3}$ ? $\newline$ 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)=(3 x-2)(x+4)(9 x+5)

has zeros at

x=-4, x=-\frac{5}{9}

, and

x=\frac{2}{3}

.

\newline

What is the sign of

f

on the interval

-\frac{5}{9}<x<\frac{2}{3}

?

\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.

Sign of $f(x)$ : Since $f(x)$ is a product of three terms, the sign of $f(x)$ depends on the sign of each term in the interval.
First term analysis: The first term $(3x-2)$ is positive when $x > \frac{2}{3}$ , and negative when $x < \frac{2}{3}$ .
Second term analysis: The second term $(x+4)$ is always positive since the zero at $x=-4$ is not in our interval.
Third term analysis: The third term $(9x+5)$ is positive when $x > -\frac{5}{9}$ , and negative when $x < -\frac{5}{9}$ .
Evaluation in interval: In the interval $-\frac{5}{9} < x < \frac{2}{3}$ , the first term $(3x-2)$ is negative, the second term $(x+4)$ is positive, and the third term $(9x+5)$ is positive.
Final result: Multiplying a negative by two positives gives a negative, so $f$ is always negative on the interval.

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