If the function f(x) = kx^3 - 9x^2 + 9x + 3 is monotonically incr… - Vidyadip

MCQ

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

A
$\text{k} < 3\text{k} < 3$
B
$\text{k}\leq3\text{k}\leq3$
✓
$\text{k} > 3\text{k} > 3$
D
$\text{k}\geq3$

✓

Answer

Correct option: C.

$\text{k} > 3\text{k} > 3$

$f(x)=k x^3-9 x^2+9 x+3$
$f^{\prime}(x)=k x^2-27$
$=3\left(x^2-9\right)$
For $\mathrm{f}(\mathrm{x})$ to be increasing, we must have
$f^{\prime}(x) > 0$
$\Rightarrow 3\left(x^2-9\right) > 0$
$\Rightarrow\left(x^2-9\right) > 0\left[\text { Since}, 3 > 0,3\left(x^2-9\right) > 0 \Rightarrow\left(x^2-9\right) > 0\right]$
$\Rightarrow(x+3)(x-3) > 0$
$\Rightarrow x < -3 $ or $ x > 3$
$\Rightarrow|x| > 3$

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