The solution below corresponds to an inequality involving a quartic function.
Write a possible quartic polynomial inequality that matches the solution. Prove that your inequality matches the solution.
x < 4 and x > 7
midway between 4 and 7 is 11/2
Since f(x) = x^4 > 0 is satisfied by all x<0 or x>0, just shift that
a simple quartic shifted down would be
x^4 - (11/2)^4 > 0
a more involved one shifted right is
(x - 11/2)^4 > 0
16x^4 - 352x^3 + 2904x^2 - 10648x + 14641 > 0
how did u get 11/2
To write a quartic polynomial inequality that matches the solution x < 4 and x > 7, we need to create a polynomial expression that satisfies these conditions.
Let's start by creating two quadratic polynomials that correspond to each inequality:
1. x < 4: We want the expression to be less than 4. We can have a quadratic equation in the form of (x - a)(x - b) < 0, where a and b are the roots. To make it quartic, we can include another pair of roots c and d. Let's choose the roots to be 3 and 2, so the quadratic polynomial becomes (x - 3)(x - 2).
2. x > 7: We want the expression to be greater than 7. Similar to the previous case, we can have another quadratic equation in the form (x - e)(x - f) > 0, where e and f are the roots. Let's choose the roots to be 8 and 9, so the quadratic polynomial becomes (x - 8)(x - 9).
Now, let's combine these two inequalities to create a quartic polynomial inequality. We need the expression to be less than 4 and greater than 7 simultaneously. To achieve this, we can multiply both polynomials:
(x - 3)(x - 2)(x - 8)(x - 9) < 0
This inequality represents a quartic polynomial that matches the given solution.
To prove that this inequality matches the solution, we can use the concept of intervals. We need to find the intervals where the expression (x - 3)(x - 2)(x - 8)(x - 9) is negative.
1. Calculate the critical points:
Setting each factor equal to zero, we find the critical points as:
x = 3, 2, 8, 9
2. Plot the critical points on a number line:
On a number line, mark the critical points at 2, 3, 8, and 9.
3. Determine the sign of the polynomial expression in each interval:
- In the interval (-∞, 2), all factors are negative. So the expression is negative.
- In the interval (2, 3), the factors (x - 2) and (x - 3) are positive, while (x - 8) and (x - 9) remain negative. So the expression is positive.
- In the interval (3, 8), all factors are positive. So the expression is positive.
- In the interval (8, 9), the factors (x - 8) and (x - 9) are positive, while (x - 2) and (x - 3) remain negative. So the expression is negative.
- In the interval (9, ∞), all factors are positive. So the expression is positive.
From the analysis above, we see that the sign of the expression (x - 3)(x - 2)(x - 8)(x - 9) follows the pattern: - + + - +. Hence, the inequality (x - 3)(x - 2)(x - 8)(x - 9) < 0 matches the given solution x < 4 and x > 7.