$f(x)=(2 x+5)(x+3)(3 x-10)(2 x-8)$ has zeros at $x=-3$, $x=-\frac{5}{2}, x=\frac{10}{3}$, and $x=4$. What is the sign of $f$ on the interval $-\frac{5}{2}
Answer (1)
Identify the zeros of f ( x ) within the interval ( − 2 5 , 4 ) : x = − 3 and x = 3 10 .
Test the sign of f ( x ) in the subintervals ( − 2 5 , − 3 ) , ( − 3 , 3 10 ) , and ( 3 10 , 4 ) using test values x = − 2.75 , x = 0 , and x = 3.5 , respectively.
Determine that f ( − 2.75 ) < 0 , 0"> f ( 0 ) > 0 , and f ( 3.5 ) < 0 .
Conclude that f ( x ) is sometimes positive and sometimes negative on the interval ( − 2 5 , 4 ) , so the answer is f is sometimes positive and sometimes negative on the interval.
Explanation
Understanding the Problem We are given the function f ( x ) = ( 2 x + 5 ) ( x + 3 ) ( 3 x − 10 ) ( 2 x − 8 ) and its zeros at x = − 3 , x = − 2 5 , x = 3 10 , and x = 4 . We want to determine the sign of f ( x ) on the interval − 2 5 0 .
So, f ( x ) is positive on the interval ( − 3 , 3 10 ) .
For the interval ( 3 10 , 4 ) , let's choose x = 3.5 . Then
f ( 3.5 ) = ( 2 ( 3.5 ) + 5 ) ( 3.5 + 3 ) ( 3 ( 3.5 ) − 10 ) ( 2 ( 3.5 ) − 8 ) = ( 12 ) ( 6.5 ) ( 0.5 ) ( − 1 ) = − 39 < 0 .
So, f ( x ) is negative on the interval ( 3 10 , 4 ) .
Determining the Sign of f(x) Since f ( x ) is negative on ( − 2 5 , − 3 ) , positive on ( − 3 , 3 10 ) , and negative on ( 3 10 , 4 ) , the sign of f ( x ) changes on the interval ( − 2 5 , 4 ) . Therefore, f ( x ) is sometimes positive and sometimes negative on the interval ( − 2 5 , 4 ) .
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 positive or negative tells us whether the object is moving forward or backward. Similarly, in economics, if f ( x ) represents the profit of a company, knowing the intervals where f ( x ) is positive helps identify periods of profitability, while negative intervals indicate losses. This type of analysis is also fundamental in optimization problems, where we seek to maximize or minimize a function.
