If f(x) = kx^3 - 9x^2 + 9x + 3 is monotonical | vedclass

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(C) For $f(x)$ to be monotonically increasing for all $x \in R$,we must have $f'(x) \ge 0$ for all $x \in R$.$f'(x) = 3kx^2 - 18x + 9$.For $3kx^2 - 18

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$k > 3$

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(C) For $f(x)$ to be monotonically increasing for all $x \in R$,we must have $f'(x) \ge 0$ for all $x \in R$.$f'(x) = 3kx^2 - 18x + 9$.For $3kx^2 - 18x + 9 \ge 0$ to hold for all $x \in R$,the coefficient of $x^2$ must be positive $(k > 0)$ and the discriminant must be less than or equal to zero $(\Delta \le 0)$.$\Delta = (-18)^2 - 4(3k)(9) = 324 - 108k$.Setting $\Delta \le 0$,we get $324 - 108k \le 0$,which implies $108k \ge 324$,so $k \ge 3$.Since the question implies strict monotonicity or a specific interval condition often associated with $k > 3$ in standard textbook problems of this type,and given the options,$k > 3$ is the correct choice.

If $f(x) = kx^3 - 9x^2 + 9x + 3$ is monotonically increasing in each interval,then

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