If `f(x) = kx^(3) -9x^(2) + 9x +3` is monotonically increasing in every interval, then which one of the following is correct?

Options

A. (A)`k lt 3`
B. (B)`k le3`
C. (C)`k gt 3`
D. (D)`k ge 3`

Answer

To determine the value of \( k \) for which the function \( f(x) = kx^3 - 9x^2 + 9x + 3 \) is monotonically increasing in every interval, we need to follow these steps:

### Step 1: Find the derivative of \( f(x) \)
The first step is to compute the derivative \( f'(x) \):

\[ f'(x) = \frac{d}{dx}(kx^3 - 9x^2 + 9x + 3) \]

Using the power rule, we differentiate each term:

\[ f'(x) = 3kx^2 - 18x + 9 \]

### Step 2: Set the derivative greater than zero
For the function to be monotonically increasing, the derivative must be greater than or equal to zero for all \( x \):

\[ f'(x) > 0 \]

This implies:

\[ 3kx^2 - 18x + 9 > 0 \]

### Step 3: Simplify the inequality
We can simplify this inequality by dividing through by 3:

\[ kx^2 - 6x + 3 > 0 \]

### Step 4: Analyze the quadratic inequality
The quadratic \( kx^2 - 6x + 3 \) must be positive for all values of \( x \). For a quadratic function \( ax^2 + bx + c \) to be positive for all \( x \), the following conditions must be satisfied:

1. The leading coefficient \( a \) (which is \( k \) here) must be greater than 0.
2. The discriminant must be less than 0.

### Step 5: Calculate the discriminant
The discriminant \( D \) of the quadratic \( kx^2 - 6x + 3 \) is given by:

\[ D = b^2 - 4ac = (-6)^2 - 4(k)(3) = 36 - 12k \]

For the quadratic to be positive for all \( x \), we need:

\[ D < 0 \]

This gives us the inequality:

\[ 36 - 12k < 0 \]

### Step 6: Solve the inequality
Now, we solve for \( k \):

\[ 36 < 12k \]
\[ k > 3 \]

### Conclusion
Thus, for the function \( f(x) \) to be monotonically increasing in every interval, the value of \( k \) must be greater than 3.

### Final Answer
The correct option is \( k > 3 \).

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