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)

Question

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.

Bytelearn · Accepted Answer

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 > (2)/(3), and negative when x < (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 > -(5)/(9), and negative when x < -(5)/(9).Evaluation in interval: In the interval -(5)/(9) < x < (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.

$f(x)=(3 x-2)(x+4)(9 x+5)$ 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}

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$f(x)=(3 x-2)(x+4)(9 x+5)$ 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}