$f(x)=(5 x+1)(4 x-8)(x+6)$ has zeros at $x=-6, x=-\frac{1}{5}$, and $x=2$. What is the sign of $f$ on the interval $-\frac{1}{5} < x < 2$?
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.
Answer (2)
Choose a test value within the interval ( − 5 1 , 2 ) , such as x = 0 .
Evaluate the function at the test value: f ( 0 ) = ( 5 ( 0 ) + 1 ) ( 4 ( 0 ) − 8 ) ( 0 + 6 ) = ( 1 ) ( − 8 ) ( 6 ) = − 48 .
Determine the sign of f ( 0 ) : Since f ( 0 ) = − 48 < 0 , the function is negative at x = 0 .
Conclude that f ( x ) is always negative on the interval ( − 5 1 , 2 ) : f is always negative on the interval.
Explanation
Understanding the Problem We are given the function f ( x ) = ( 5 x + 1 ) ( 4 x − 8 ) ( x + 6 ) and its zeros at x = − 6 , x = − 5 1 , and x = 2 . We want to determine the sign of f ( x ) on the interval − 5 1 < x < 2 .
Choosing a Test Value To determine the sign of f ( x ) on the interval ( − 5 1 , 2 ) , we can choose a test value x 0 in this interval and evaluate f ( x 0 ) . If 0"> f ( x 0 ) > 0 , then f ( x ) is positive on the interval, and if f ( x 0 ) < 0 , then f ( x ) is negative on the interval. Since f ( x ) is a polynomial, it is continuous, and thus the sign of f ( x ) will be the same throughout the interval ( − 5 1 , 2 ) .
Evaluating f(0) Let's choose x 0 = 0 as our test value, which lies in the interval ( − 5 1 , 2 ) . Now we evaluate f ( 0 ) :
f ( 0 ) = ( 5 ( 0 ) + 1 ) ( 4 ( 0 ) − 8 ) ( 0 + 6 ) = ( 1 ) ( − 8 ) ( 6 ) = − 48 .
Determining the Sign Since f ( 0 ) = − 48 < 0 , the function f ( x ) is negative on the interval ( − 5 1 , 2 ) .
Analyzing the Factors Alternatively, we can analyze the sign of each factor in the interval ( − 5 1 , 2 ) .
The sign of 5 x + 1 is positive since -\frac{1}{5}"> x > − 5 1 .
The sign of 4 x − 8 is negative since x < 2 .
The sign of x + 6 is positive since -6"> x > − 6 .
Therefore, the sign of f ( x ) is ( + ) ( − ) ( + ) = ( − ) .
Final Answer Thus, f ( x ) is always negative on the interval ( − 5 1 , 2 ) .
Examples
Understanding the sign of a function over an interval is crucial in many real-world applications. For instance, in physics, if f ( x ) represents the velocity of an object, knowing where f ( x ) is negative tells us the object is moving in the opposite direction. Similarly, in economics, if f ( x ) represents profit, knowing the intervals where f ( x ) is positive indicates profitable production levels. This concept is also fundamental in optimization problems, where we seek to maximize or minimize a function by analyzing its increasing and decreasing intervals.
The function f ( x ) = ( 5 x + 1 ) ( 4 x − 8 ) ( x + 6 ) is negative on the interval − 5 1 < x < 2 . This was confirmed by testing a value in the interval, which resulted in a negative function value. Thus, the chosen option is that f is always negative on the interval.
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