If the function f(x) = kx3 - 9x2 + 9 x + 3 is monotonically increasing in every interval, then

To determine if the function is monotonically increasing in every interval, we need to find its derivative and set it greater than zero.

The derivative of the function f(x) = kx^3 - 9x^2 + 9x + 3 is f'(x) = 3kx^2 - 18x + 9.

Setting f'(x) > 0 gives us 3kx^2 - 18x + 9 > 0.

This is a quadratic inequality. To solve it, we first find the roots of the equation 3kx^2 - 18x + 9 = 0.

Using the quadratic formula, the roots are x = [18 ± sqrt((18)^2 - 43k9)] / (2*3k) = [18 ± sqrt(324 - 108k)] / 6k.

For the function to be monotonically increasing, the discriminant (the value under the square root) must be less than or equal to zero. This is because if the discriminant is greater than zero, the quadratic equation will have two distinct real roots, meaning the function will decrease between these roots.

So, we have 324 - 108k ≤ 0.

Solving this inequality gives us k ≥ 3.

Therefore, for the function f(x) = kx^3 - 9x^2 + 9x + 3 to be monotonically increasing in every interval, k must be greater than or equal to 3.