The set of all positive real values of k, for which the equation `x^(3) - 9x^(2) + 24x - k = 0` has three distinct real roots, is the interval :

Options

A. (18, 21)
B. (16,20)
C. (14,18)
D. 12, 16)

Answer

To solve the problem of finding the set of all positive real values of \( k \) for which the equation

\[ x^3 - 9x^2 + 24x - k = 0 \]

has three distinct real roots, we can follow these steps:

### Step 1: Define the function
Let

\[ f(x) = x^3 - 9x^2 + 24x - k \]

We need to find conditions on \( k \) such that \( f(x) = 0 \) has three distinct real roots.

### Step 2: Find the derivative
To determine the critical points, we first find the derivative of \( f(x) \):

\[ f'(x) = 3x^2 - 18x + 24 \]

### Step 3: Set the derivative to zero
We set the derivative equal to zero to find the critical points:

\[ 3x^2 - 18x + 24 = 0 \]

Dividing the entire equation by 3 gives:

\[ x^2 - 6x + 8 = 0 \]

### Step 4: Factor the quadratic
Now we can factor the quadratic:

\[ (x - 2)(x - 4) = 0 \]

Thus, the critical points are:

\[ x = 2 \quad \text{and} \quad x = 4 \]

### Step 5: Determine the nature of critical points
Next, we check the second derivative to determine whether these critical points are maxima or minima:

\[ f''(x) = 6x - 18 \]

Calculating the second derivative at the critical points:

- For \( x = 2 \):

\[ f''(2) = 6(2) - 18 = 12 - 18 = -6 \quad (\text{local maximum}) \]

- For \( x = 4 \):

\[ f''(4) = 6(4) - 18 = 24 - 18 = 6 \quad (\text{local minimum}) \]

### Step 6: Find the values of \( f \) at the critical points
Now we evaluate \( f(x) \) at the critical points to find the corresponding \( k \) values:

- For \( x = 2 \):

\[ f(2) = 2^3 - 9(2^2) + 24(2) - k = 8 - 36 + 48 - k = 20 - k \]

- For \( x = 4 \):

\[ f(4) = 4^3 - 9(4^2) + 24(4) - k = 64 - 144 + 96 - k = 16 - k \]

### Step 7: Set conditions for distinct roots
For \( f(x) \) to have three distinct real roots, we need:

1. \( f(2) > 0 \) (local maximum)
2. \( f(4) < 0 \) (local minimum)

This gives us the inequalities:

1. \( 20 - k > 0 \) → \( k < 20 \)
2. \( 16 - k < 0 \) → \( k > 16 \)

### Step 8: Combine the inequalities
Combining these inequalities, we find:

\[ 16 < k < 20 \]

### Conclusion
Thus, the set of all positive real values of \( k \) for which the equation has three distinct real roots is the interval:

\[ (16, 20) \]