f(x) = (x+4)(x-2)(x-1) has zeros at x=-4, x=1, and x=2. What is the sign of f on the interval -4<x<1?
To determine the sign of f(x) on the interval -4 < x < 1, we need to examine the intervals defined by the zeros of f(x) and consider the sign of f(x) within each interval.
We know that f(x) = (x+4)(x-2)(x-1). Let's look at each factor individually:
(x+4) changes sign at x = -4.
(x-2) changes sign at x = 2.
(x-1) changes sign at x = 1.
Now, let's consider the intervals formed by these zeros:
-∞ < x < -4
-4 < x < 1
1 < x < 2
2 < x < ∞
On the interval -4 < x < 1, both (x+4) and (x-1) are negative, while (x-2) is positive.
So, f(x) = (x+4)(x-2)(x-1) will be negative on the interval -4 < x < 1.
To determine the sign of f(x) on the interval -4 < x < 1, we need to evaluate f(x) at a test point within that interval. Let's choose x = 0 as our test point.
Now, substitute x = 0 into the equation f(x) = (x+4)(x-2)(x-1):
f(0) = (0+4)(0-2)(0-1)
= (4)(-2)(-1)
= -8
The sign of f(x) at x = 0, which lies in the interval -4 < x < 1, is negative (-).
Therefore, the sign of f(x) on the interval -4 < x < 1 is negative (-).
To determine the sign of f(x) on the interval -4 < x < 1, we need to evaluate f(x) for a value of x within that interval and analyze the result.
Let's choose a test value, say x = 0, which lies within the interval -4 < x < 1. We'll substitute x = 0 into the function f(x) = (x+4)(x-2)(x-1) and observe the sign of the resulting expression.
f(0) = (0+4)(0-2)(0-1) = 4 * (-2) * (-1) = -8
Since f(0) = -8, the function f(x) is negative on the interval -4 < x < 1.
Therefore, the sign of f(x) on the interval -4 < x < 1 is negative.