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  4. The function f(x) = Kx3 - 9x2 + 9x + 3 is monotonically increasing in each interval, then

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Q. The function $f(x) = Kx^3 - 9x^2 + 9x + 3$ is monotonically increasing in each interval, then

Application of Derivatives

Solution:

$f'(x) = 3\, Kx^2 - 18x + 9$
$ = 3 (Kx^2 - 6x + 3)$
Since $f(x)$ is monotonically increasing
$\therefore f'\left(x\right) \ge 0$
$\therefore Kx^{2}-6r + 3 \ge 0\,\forall\, x \in R$
$\therefore \Delta = b^{2} - 4ac < 0$, $K > 0$
i.e., $36 - 12\, K < 0$
$\Rightarrow 36 < 12 \,K$
$\Rightarrow 3 < K$
$\Rightarrow K > 3 \quad\cdot$ [so that $K > 0$]
Hence $K > 3$