Question

The function $f(x)=K{x}^3-9x^2+9x+3$ is monotonically increasing in each interval, then

• A. $K<3$
• B. $K\le 3$
• C. $K>3$
• D. none of these

Answer 1

Answer: (C)

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