The maximum value of the function `f(x)=3x^(3)-18x^(2)+27x-40` on the set `S={x in R: x^(2)+30 le 11x}` is:

Options

A. `-122`
B. `-222`
C. `222`
D. `122`

Answer

To find the maximum value of the function \( f(x) = 3x^3 - 18x^2 + 27x - 40 \) on the set \( S = \{ x \in \mathbb{R} : x^2 + 30 \leq 11x \} \), we will follow these steps:

### Step 1: Determine the set \( S \)

We start with the inequality:

\[ x^2 + 30 \leq 11x \]

Rearranging gives:

\[ x^2 - 11x + 30 \leq 0 \]

### Step 2: Factor the quadratic

Next, we factor the quadratic expression:

\[ x^2 - 11x + 30 = (x - 5)(x - 6) \leq 0 \]

### Step 3: Find the roots

The roots of the equation \( (x - 5)(x - 6) = 0 \) are \( x = 5 \) and \( x = 6 \).

### Step 4: Analyze the sign of the quadratic

We analyze the sign of \( (x - 5)(x - 6) \) on the number line:

- For \( x < 5 \): The expression is positive.
- For \( 5 \leq x \leq 6 \): The expression is negative or zero.
- For \( x > 6 \): The expression is positive.

Thus, the set \( S \) is:

\[ S = [5, 6] \]

### Step 5: Differentiate the function \( f(x) \)

Now we differentiate \( f(x) \):

\[ f'(x) = 9x^2 - 36x + 27 \]

### Step 6: Factor the derivative

We can factor \( f'(x) \):

\[ f'(x) = 9(x^2 - 4x + 3) = 9(x - 1)(x - 3) \]

### Step 7: Find critical points

The critical points are \( x = 1 \) and \( x = 3 \). We need to check the behavior of \( f'(x) \) in the intervals defined by these points.

### Step 8: Analyze the intervals

- For \( x < 1 \): \( f'(x) > 0 \) (increasing)
- For \( 1 < x < 3 \): \( f'(x) < 0 \) (decreasing)
- For \( x > 3 \): \( f'(x) > 0 \) (increasing)

### Step 9: Determine maximum in the interval \( S \)

Since \( S = [5, 6] \) and \( f'(x) \) is increasing in this interval, the maximum value occurs at the endpoint \( x = 6 \).

### Step 10: Calculate \( f(6) \)

Now we calculate \( f(6) \):

\[ f(6) = 3(6^3) - 18(6^2) + 27(6) - 40 \]

Calculating each term:

\[ = 3(216) - 18(36) + 162 - 40 \]
\[ = 648 - 648 + 162 - 40 \]
\[ = 162 - 40 = 122 \]

### Conclusion

The maximum value of the function \( f(x) \) on the set \( S \) is:

\[ \boxed{122} \]