If `f(x)=k x^3-9x^2+9x+3` monotonically increasing in `R ,` then

Options

A. `klt3`
B. `kle2`
C. `kge3`
D. none of these

Explanation

To determine the condition under which the function \( f(x) = kx^3 - 9x^2 + 9x + 3 \) is monotonically increasing for all real numbers \( x \), we need to analyze the first derivative of the function.

### Step 1: Find the first derivative \( f'(x) \)

The first derivative of \( f(x) \) is calculated as follows:

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

Using the power rule of differentiation:

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

### Step 2: Set the first derivative greater than or equal to zero

For the function to be monotonically increasing, we need:

\[ f'(x) \geq 0 \]

This means:

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

### Step 3: Simplify the inequality

We can factor out the common term from the inequality:

\[ 3(kx^2 - 6x + 3) \geq 0 \]

This simplifies to:

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

### Step 4: Analyze the quadratic expression

The expression \( kx^2 - 6x + 3 \) is a quadratic function. For this quadratic to be non-negative for all \( x \), it must either open upwards (which occurs when \( k > 0 \)) and have a non-positive discriminant.

### Step 5: Calculate the discriminant

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

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

### Step 6: Set the discriminant less than or equal to zero

To ensure that the quadratic does not have any real roots (which would mean it does not cross the x-axis), we need:

\[ D \leq 0 \]

This gives us:

\[ 36 - 12k \leq 0 \]

### Step 7: Solve for \( k \)

Rearranging the inequality:

\[ 36 \leq 12k \]

Dividing both sides by 12:

\[ 3 \leq k \]

Or equivalently:

\[ k \geq 3 \]

### Conclusion

Thus, for the function \( f(x) = kx^3 - 9x^2 + 9x + 3 \) to be monotonically increasing for all real \( x \), the condition is:

\[ k \geq 3 \]