$f(x)=(5 x+3)(x-2)(3 x+7)(x+5)$ has zeros at $x=-5$, $x=-\frac{7}{3}, x=-\frac{3}{5}$, and $x=2$. What is the sign of $f$ on the interval $-\frac{7}{3}
Answer (1)
Choose a test value within the interval: x = − 2 .
Evaluate the function at the test value: f ( − 2 ) = ( 5 ( − 2 ) + 3 ) (( − 2 ) − 2 ) ( 3 ( − 2 ) + 7 ) (( − 2 ) + 5 ) = 84 .
Determine the sign: Since 0"> f ( − 2 ) = 84 > 0 , the function is positive at x = − 2 .
Conclude the sign of f ( x ) on the interval: f is always positive on the interval − 3 7 0 , then f ( x ) is positive on the interval, and if f ( x 0 ) < 0 , then f ( x ) is negative on the interval. A convenient choice for x 0 is x 0 = − 2 , since − 3 7 ≈ − 2.33 and − 5 3 = − 0.6 , so − 2 lies in the interval ( − 3 7 , − 5 3 ) .
Evaluating the Function Now we evaluate f ( − 2 ) :
f ( − 2 ) = ( 5 ( − 2 ) + 3 ) (( − 2 ) − 2 ) ( 3 ( − 2 ) + 7 ) (( − 2 ) + 5 ) = ( − 10 + 3 ) ( − 4 ) ( − 6 + 7 ) ( − 2 + 5 ) = ( − 7 ) ( − 4 ) ( 1 ) ( 3 ) = 84 .
Since 0"> f ( − 2 ) = 84 > 0 , the function is positive at x = − 2 .
Determining the Sign Since 0"> f ( − 2 ) > 0 and there are no zeros of f ( x ) in the interval ( − 3 7 , − 5 3 ) , the sign of f ( x ) remains constant on this interval. Therefore, f ( x ) is always positive on the interval ( − 3 7 , − 5 3 ) .
Final Answer Therefore, the sign of f on the interval $-\frac{7}{3}
