To find the maximum value of a and the minimum value of b, we need to solve the equation 9+9x-x²-x³ = k for x and then determine the range of values for k where the equation has only one solution.
First, we can rearrange the equation to get:
x³ + x² - 9x - (k - 9) = 0
We can then use the rational root theorem to find the possible rational roots of this polynomial equation. The possible rational roots are of the form p/q, where p is a factor of the constant term (k - 9) and q is a factor of the leading coefficient (1). The factors of (k - 9) are ±1, ±3, ±(k-9), and the factors of 1 are ±1. So the possible rational roots are:
±1, ±3, ±(k-9), ±(k-9)/3
We can test each of these possible roots by plugging them into the equation and checking if the result is zero. After testing each possible root, we find that the only root that works is x = 3. Therefore, the equation has only one solution when x = 3.
To determine the range of values for k where the equation has only one solution, we can substitute x = 3 into the equation:
9 + 9(3) - 3² - 3³ = k
Simplifying this, we get:
k = -9
So the equation has only one solution when k = -9. Now, we need to find the maximum value of a and the minimum value of b for this value of k.
For k < -9, the equation has no real solutions, since the equation is a cubic polynomial with a negative leading coefficient, which means it has a local maximum and a local minimum. Therefore, the maximum value of a is -9.
For k > -9, the equation has three real solutions, since the equation is a cubic polynomial with a negative leading coefficient, which means it has a local maximum and a local minimum, and crosses the x-axis three times. Therefore, the minimum value of b is -9.
So the maximum value of a is -9 and the minimum value of b is -9.