f(x)=(2x-3)(x+6)(5x+6) has zeros at x=-6,x=-(6)/(5), and x=(3)/(2). What is the sign of f on the interval -6 < x < -(6)/(5) ? 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.

Identify Sign Changes: Since f(x) has zeros at x=-6, x=-(6)/(5), and x=(3)/(2), we know that the function changes sign at each of these points.Test Interval: We need to test a value between -6 and -(6)/(5) to determine the sign of f(x) on that interval. Let's pick x=-5.5, which is between -6 and -(6)/(5).Calculate f(-5.5): Plug x=-5.5 into f(x) to see the sign: f(-5.5)=(2(-5.5)-3)((-5.5)+6)(5(-5.5)+6).Calculate each term: (2(-5.5)-3) = -11-3 = -14, ((-5.5)+6) = 0.5, (5(-5.5)+6) = -27.5+6 = -21.5.

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f(x)=(2x-3)(x+6)(5x+6) has zeros at
x=-6,x=-(6)/(5), and
x=(3)/(2).
What is the sign of
f on the interval
-6 < x < -(6)/(5) ?
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.

$f(x)=(2 x-3)(x+6)(5 x+6)$ has zeros at $x=-6, x=-\frac{6}{5}$ , and $x=\frac{3}{2}$ . $\newline$ What is the sign of $f$ on the interval \( -6

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Precalculus

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Full solution

f(x)=(2 x-3)(x+6)(5 x+6)

has zeros at

x=-6, x=-\frac{6}{5}

, and

x=\frac{3}{2}

\newline

What is the sign of

f

on the interval

-6<x<-\frac{6}{5}

\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 Sign Changes: Since $f(x)$ has zeros at $x=-6$ , $x=-\frac{6}{5}$ , and $x=\frac{3}{2}$ , we know that the function changes sign at each of these points.
Test Interval: We need to test a value between $-6$ and $-\frac{6}{5}$ to determine the sign of $f(x)$ on that interval. Let's pick $x=-5.5$ , which is between $-6$ and $-\frac{6}{5}$ .
Calculate $f(-5.5)$ : Plug $x=-5.5$ into $f(x)$ to see the sign: $f(-5.5)=(2(-5.5)-3)((-5.5)+6)(5(-5.5)+6)$ .
Calculate $f(-5.5)$ : Plug $x=-5.5$ into $f(x)$ to see the sign: $f(-5.5)=(2(-5.5)-3)((-5.5)+6)(5(-5.5)+6)$ .Calculate each term: $(2(-5.5)-3) = -11-3 = -14$ , $((-5.5)+6) = 0.5$ , $(5(-5.5)+6) = -27.5+6 = -21.5$ .