Determine the intervals on which f(x) is positive and on which f(x) is negative.
f(x) = x^4+2x^3-13x^2-14x+24
f(x) < 0 if _____________________
f(x) > 0 if ____________________
f(x) = (x+4)(x+2)(x-1)(x-3)
f(x) changes sign where it crosses the x-axis.
Since f(x) has even degree, with a positive leading coefficient, you know that f is positive for x < -4, since all the factors are positive.
So, f(x) is
positive on (-∞,-4)
negative on (-4,-2)
positive on (-2,1)
negative on (1,3)
positive on (3,∞)
f(x) < 0 if the equation is below the x-axis, meaning the function is negative.
f(x) > 0 if the equation is above the x-axis, meaning the function is positive.
To determine the intervals on which f(x) is positive and negative, we can find the x-intercepts of the function. By setting f(x) equal to zero and solving for x, we can find the x-values where the function crosses the x-axis.
Setting f(x) = x^4+2x^3-13x^2-14x+24 equal to zero:
x^4+2x^3-13x^2-14x+24 = 0
Using a graphing calculator or factoring, we find the x-values for which the function crosses the x-axis:
x ≈ -3.36, -0.95, 1.58, and 2.73
Now, we can determine the intervals on which f(x) is positive and negative by analyzing the behavior of the function between these x-values.
Let's also analyze the sign of the different terms of the function:
- x^4: Negative for all x
- 2x^3: Positive for all x
- -13x^2: Negative for x between -3.36 and -0.95, and positive for x between -0.95 and 1.58, negative again for x between 1.58 and 2.73, and positive for all x greater than 2.73
- -14x: Negative for x between -3.36 and -0.95, positive for x between -0.95 and 1.58, negative for x between 1.58 and 2.73, and positive for all x greater than 2.73
- 24: Positive for all x
Based on this analysis, we can conclude:
f(x) < 0 if x is between -3.36 and -0.95, and x is between 1.58 and 2.73.
f(x) > 0 if x is less than -3.36 or between -0.95 and 1.58, or x is greater than 2.73.
Remember, this analysis is based on the behavior of the function between the x-intercepts, and it assumes the function is continuous between these points.
To determine the intervals on which f(x) is positive and negative, we need to find the x-values where the function crosses the x-axis (f(x) = 0) and use that information to determine the sign of f(x) in each interval.
First, find the x-intercepts by setting f(x) equal to zero:
x^4+2x^3-13x^2-14x+24 = 0
From there, you can either factor the equation or use a numerical method such as the rational root theorem or a graphing calculator to find the x-intercepts.
By factoring, the equation can be rewritten as:
(x+2)(x-2)(x+3)(x-1) = 0
Solving for x, we have the following x-intercepts:
x = -2, 2, -3, 1
Now we can construct a sign chart to determine the intervals on which f(x) is positive or negative:
-3 -2 1 2
+ - + -
Using the sign chart, we can determine the intervals where f(x) < 0 (negative) and f(x) > 0 (positive):
f(x) < 0 if x ∈ (-3, -2) U (1, 2)
f(x) > 0 if x ∈ (-∞, -3) U (-2, 1) U (2, ∞)
Therefore, the function f(x) is positive on the intervals (-∞, -3) U (-2, 1) U (2, ∞) and negative on the intervals (-3, -2) U (1, 2).
To determine the intervals on which f(x) is positive (f(x) > 0) and on which f(x) is negative (f(x) < 0), we need to find the roots of the given function.
Step 1: Find the critical points by finding the roots of f(x).
Set f(x) = 0 and solve for x.
x^4+2x^3-13x^2-14x+24 = 0
You can use various methods to solve this equation, such as factoring, using the rational root theorem, or graphing to find the roots. In this case, the equation does not appear to factor easily and the rational root theorem does not provide any rational roots, so graphing or using a calculator may be the most efficient option.
Using a graphing calculator or software, plot the function f(x) = x^4+2x^3-13x^2-14x+24 and find the x-values where the function crosses the x-axis (where f(x) = 0).
After finding the roots of the equation, let's say x1, x2, x3, and x4, we can proceed with the next step.
Step 2: Analyze the intervals between the roots.
To determine the intervals on which f(x) is positive or negative, we can create a sign chart based on the roots we found. A sign chart helps us determine the sign of the function in different intervals.
Write down the roots x1, x2, x3, x4 in increasing order from left to right on the number line.
x1 x2 x3 x4
Now, pick a test value from each interval:
- Choose a test value less than the smallest root and substitute it into the equation f(x).
- Choose a test value between each pair of adjacent roots and substitute it into the equation f(x).
- Choose a test value greater than the largest root and substitute it into the equation f(x).
Evaluate f(test value) for each interval and determine the sign of f(x) in that interval.
For example, if the results are:
f(test value less than x1) > 0
f(test value between x1 and x2) < 0
f(test value between x2 and x3) > 0
f(test value greater than x3) < 0
Then, the intervals on which f(x) is positive and negative are:
f(x) > 0 if x < x1 or x2 < x < x3
f(x) < 0 if x1 < x < x2 or x3 < x
Note: In some cases, the function may touch or cross the x-axis at the root points, but if it does not change signs at these points, we do not include them as part of the intervals where f(x) is positive or negative.
By following these steps, you can determine the intervals on which f(x) is positive (f(x) > 0) and on which f(x) is negative (f(x) < 0) using the given function f(x) = x^4+2x^3-13x^2-14x+24.