Question

Q.2

Calculate the x-values of the turning points of y=9+9x-r-r³.
Show all your working and give your answers correct to 2 decimal places.
The equation 9+9x-x²-x³ = k has one solution only when k < a and when k > b,
where a and b are integers.
Find the maximum value of a and the minimum value of b.

Answer 1

To find the turning points of y = 9 + 9x - x² - x³, its derivative is calculated, set to zero, and solved for x. The maximum value of a and minimum value of b are determined by solving for the critical value of k when the discriminant of the derivative is zero.

Step-by-step explanation:

Firstly, to find the turning points of the function y = 9 + 9x - x² - x³, the derivative of y with respect to x should be taken and then set to zero.

The derivative is: y' = 9 - 2x - 3x². To find the turning points, we solve y' = 0. This gives a quadratic equation in x, which can be solved using the quadratic formula: x = [-(-2) ± √((-2)² - 4(-3)(9))]/(2(-3)). The x-values of the turning points can then be rounded to two decimal places.

To determine the values of a and b for which the equation 9+9x-x²-x³ = k has exactly one solution, we need to look at the discriminant of the derivative of y, which is Δ = (-2)² - 4(-3)(9). A quadratic equation has one real solution when its discriminant is zero. Therefore, we solve for k, when Δ = 0. This will provide the critical value of k at which the number of solutions changes, thus giving us the maximum value of a and minimum value of b.

Answer 2

Step-by-step explanation:

To find the x-values of the turning points of the function y = 9 + 9x - r - r³, we need to find the critical points of the function. The critical points occur where the derivative of the function is equal to zero.

Let's first find the derivative of the function:

y = 9 + 9x - r - r³

y' = 9 - 3r²

Setting y' equal to zero and solving for r:

9 - 3r² = 0

3r² = 9

r² = 3

r = ±√3

Now that we have the values of r, we can substitute them back into the original equation to find the corresponding x-values of the turning points:

For r = √3:

y = 9 + 9x - √3 - (√3)³

y = 9 + 9x - √3 - 3√3

y = 9 + 9x - 4√3

For r = -√3:

y = 9 + 9x + √3 - (√3)³

y = 9 + 9x + √3 + 3√3

y = 9 + 9x + 4√3

Now, let's set y equal to zero and solve for x:

For r = √3:

9 + 9x - 4√3 = 0

9x = -9 + 4√3

x = (-9 + 4√3)/9 ≈ -0.38

For r = -√3:

9 + 9x + 4√3 = 0

9x = -9 - 4√3

x = (-9 - 4√3)/9 ≈ -1.29

Therefore, the x-values of the turning points are approximately -0.38 and -1.29.

Now, let's move on to finding the maximum value of a and the minimum value of b for the equation 9 + 9x - x² - x³ = k.

To determine the maximum value of a, we need to find the largest value of k such that the equation has only one solution. This occurs when the discriminant of the equation, b² - 4ac, is less than zero.

The equation 9 + 9x - x² - x³ = k can be rewritten as -x³ - x² + 9x + (9 - k) = 0.

Comparing this equation to the standard form ax³ + bx² + cx + d = 0, we have:

a = -1

b = -1

c = 9

d = 9 - k

The discriminant is given by:

D = b² - 4ac

Substituting the values:

D = (-1)² - 4(-1)(9)

D = 1 + 36

D = 37

Since D is positive (greater than zero), the equation has three real roots for all values of k. Therefore, there is no maximum value of a.

To find the minimum value of b, we need to determine the range of k values for which the equation has three real roots. Since there is no maximum value of a, the equation has three real roots for all values of k. Therefore, there is no minimum value of b.

In summary, the equation 9 + 9x - x² - x³ = k has three real roots for all values of k, and there is no maximum value of a or minimum